Introduction

Energy saving is an important technological topic with many awaiting challenges, in particular in the case of the development of mobile technologies integrating a growing number of functionalities. Accompanying low-energy consumption device and high-power density battery development, energy harvesting technologies aim also at increasing portable electronic device autonomy1,2,3. Temperature gradients being usually present in microelectronic setups, integration of thermoelectric (TE) devices is currently explored1,3. TE solutions based on thin films compatible with the complementary-metal–oxide–semiconductor technology (CMOS) have already been proposed supporting TE device integration4,5. However, TE solutions for microelectronic applications need to operate close to room temperature (RT), TE efficiency being mainly dependent on intrinsic material properties, such as thermal conductivity (κ), electrical conductivity (σ), and Seebeck coefficient (S), TE technology improvement requires either the development of new materials, or to develop engineering methods allowing TE properties of current materials to be improved. Concurrent with κ engineering, recent band engineering solutions were proposed to increase the TE power factor PF = S2σ6, such as modulation doping7,8, resonance levels9,10, energy filtering11,12,13, and quantum confinement14. The thermopower S is related to the Peltier coefficient П such as S = П/T, П corresponding to the energy carried by the mobile charge carriers in the material per unit of charge. Figure 1a presents the method used for the measurement of S = ΔV/ΔT. Semiconductor materials are extensively studied15,16,17,18, as a same semiconductor can be used as n-type or p-type TE material depending on doping, and they allow substantial band engineering. However, interest of spin effects on material TE properties is growing, and investigations on magnetic material potential for TE applications has considerably raised. In particular, the spin-Seebeck effect offers new routes for converting waste heat to electric power19. Based on spin transport, spin-Seebeck was demonstrated in different types of ferromagnetic materials: metallic20, semiconductor21, and insulator22. Investigations of magnetism influence on material TE performance have been reported23, and original ferromagnetic materials23,24,25 and spintronic structures24,26 have been proposed for TE energy harvesting. Spin effects on the conventional Seebeck coefficient were shown to provide interesting ways of S engineering, based on charge carrier interactions with localized magnetic moments through the magnon-drag effect27 or the spin fluctuation effect28, for example. Thus, spin effect engineering in ferromagnetic (FM) materials should be considered as a possible way of obtaining improved TE properties, and interactions of mobile charge carriers with localized magnetic moments should be thoroughly investigated.

Figure 1
figure 1

Schematics of S measurements in FM thin films and schematics of electronic DOS in PM and FM metallic materials under temperature gradient. (a) S measurement principle. (b) DOS of a PM metallic-type material under temperature gradient. (c) DOS of a FM metallic-type material under temperature gradient. (d) µI distribution in a FM thin film under temperature gradient without magnetic field, or at saturation with a parallel (H//) or perpendicular (H⊥) magnetic field.

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In this work, the influence of interactions between localized magnetic moments and spin-polarized free electrons on the spin-dependent Seebeck coefficient is investigated in FM thin films exhibiting metallic conduction and low charge carrier density. In order to separate as much as possible the effects of magnetic moment fluctuation, ordering, and density, the investigations focus on two FM germanides Mn5Ge329,30,31,32,33,34 and MnCoGe35,36,37,38,39,40,41,42,43, with magnetization (M) depending on a single element (almost single magnetic moment µI carried by Mn ions, Supplementary Fig. S1), and exhibiting Curie temperatures (Tc) close to RT, allowing S measurements and M variations at Tc to be compared in the same temperature range. The film geometry allows µI degree of freedom to be reduced, as the moments are forced to be aligned in the film plane42,44.

Spin-dependent Seebeck coefficient

In the following, the term spin-dependent Seebeck coefficient is used to designate the regular Seebeck coefficient measured in FM materials below the Curie temperature. For paramagnetic (PM) metallic materials in a temperature gradient, S depends on the asymmetry of a single density of states (DOS) g(E) close to the average Fermi level (EF) corresponding to the considered temperature gradient (Fig. 1b). The average electrochemical potential (µ1) of electrons above EF promotes the diffusion of electrons towards the sample’s cold side, while the average electrochemical potential (µ2) of electrons below EF promotes the diffusion of electrons towards the sample’s hot side (Fig. 1b), due to the difference of electronic state filling between the two sides.

The net electron flux is given by the difference between these two fluxes, and leads to the accumulation of electrons at one side of the sample and immobile matrix ions at the other side of the sample, building the potential difference ΔV =  − (µ1 − µ2)/e (e is the elementary charge). The Seebeck coefficient can be expressed as:

$$S = - {\triangle\mu /eT,}$$
(1)

with Δµ = µ1 − µ2 (Fig. 1b). However, S depends on the asymmetry of the two DOS g+(E) and g-(E) of respectively majority-spin and minority-spin electrons (Fig. 1c), leading to two separate electronic currents in the case of FM materials: (i) the current of majority-spin electrons related to g+(E) and the electron densities N+ above EF and n+ below EF, as well as (ii) the current of minority-spin electrons related to g−(E) and the electron densities N− above EF and n− below EF. In this case Δµ = Δµ+  + Δµ− in Eq. 1, with Δµ+  = µ1+  − µ2+ and Δµ−  = µ1− − µ2−. Considering the simplified model presented in Fig. 1c, S can be expressed as

$$S=-\frac{1}{2eT}\frac{d\varepsilon }{n}\left[{\left(\frac{d{g}^{+}}{d\epsilon }\right)}_{{E}_{F}}-{\left(\frac{d{g}^{-}}{d\epsilon }\right)}_{{E}_{F}}\right]\left({\Delta E}_{C}+\alpha \Delta {E}_{eI}+\Delta R{k}_{B}T\right),$$
(2)

considering that the FM material of interest reports a relatively low density of carriers45. The carrier density in the considered Mn5Ge3 and MnCoGe films is respectively 1.6 × 1020 cm−3 and 1.7 × 1018 cm−3 according to RT Hall effect measurements. dε is the energy variation around EF involved with the temperature gradient, and n = n+  + n− is the entire number of electronic state close to EF (below EF according to Fig. 1c). ΔEC = EC− − EC+ = I in Fig. 1c corresponds to the energy difference between the bottom energies EC− and EC+ of the conduction bands of the minority-spin and majority-spin electrons, respectively. ΔEeI = EeI− − EeI+ with EeIj the average energy related to the polarization of the magnetic moment of conduction electrons of spin polarization j by the localized magnetic moments. This parameter does not consider macroscopic effects, such as those due to magnetic domains’ walls for example. Thus, the parameter α is added in order to take into account a statistical efficiency of free electron polarization. ΔR = R+ − R− with Rj a dimensionless constant related to the average energy of electrons of spin polarization j depending on scattering mechanisms46,47. Considering that at given T (dg+/dε)EF = Ω+ and (dg−/dε)EF = Ω−, one can define a parameter β(T) = dε/n (Ω+ − Ω−). In this case

$$S=-\frac{\beta }{2eT}\left({\Delta E}_{C}+\alpha \Delta {E}_{eI}+\Delta R{k}_{B}T\right).$$
(3)

Localized magnetic moment fluctuation

The diffractogram (a) in Fig. 2a was acquired on the Mn5Ge3 film (Supplementary Fig. S1a). The film is polycrystalline and the Scherrer equation48 applied to the most intense diffraction peak Mn5Ge3(211) indicates that the layer is composed of columnar grains with a thickness ~ 49 ± 5 nm. Grains exhibit an average lateral size ~ 1 µm and the root mean square (RMS) surface roughness of the film is ~ 1.8 nm according to AFM measurements44. Figure 3a shows the variation of the Mn5Ge3 film magnetization versus temperature in the temperature range 175 ≤ T ≤ 350 K. M decreases as temperature increases up to the Curie temperature of the FM/PM transition at Tc = 297 K. The Curie temperature corresponds to the carbon-free Mn5Ge3 compound49. The electrical conductivity variation of the sample versus T is presented in the inset. The conductivity decreases almost linearly when the temperature increases from 150 K up to Tc. Figure 3b displays the variations of the spin-dependent S of the layer in the same temperature range. The black line corresponds to measurements performed without external magnetic field. The Mn5Ge3 film was not exposed to any magnetic field before Seebeck measurements. Consequently, the net magnetization is essentially zero in these conditions. The gray envelop around the data corresponds to the maximum measurement error observed on S in this study. This error considers both elaboration and Seebeck measurement reproducibility. It is only shown on this measurement for clarity.

Figure 2
figure 2

Microstructures of the 50 nm-thick Mn5Ge3 film and of the 150 nm-thick MnxCoyGe1−x−y films. (a) X-ray diffractograms measured on Mn5Ge3 and MnxCoyGe1−x−y films. (b) AFM measurements performed on the film MnCoGe.

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Figure 3
figure 3

Magnetization (M) and Seebeck coefficient (S) measurements performed on Mn5Ge3 and MnCoGe thin films. M (a) and S (b) of a 50 nm-thick Mn5Ge3 film as a function of temperature. The inset in (a) presents the electrical conductivity of the film versus temperature. (c) M measured at 200 K (black solid squares) and 270 K (blue dashed line) on a same Mn5Ge3 film as a function of in-plane magnetic field intensity. M (d) and S (e) of 150 nm-thick MnxCoyGe1−x−y films as a function of temperature. (f) S(T/Tc) of same MnxCoyGe1−x−y films plotted as a function of Mn composition x.

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Mn5Ge3 is n-type as S < 0. S decreases as temperature below the FM/PM transition increases, following a linear behavior from T ~ 215 K to T = Tc. Remarkably, S and σ are found to concurrently increase as temperature decreases, leading to a PF increase > 1700% from 300 to 175 K. Without external field, the Mn5Ge3 film exhibits weak magnetization due to moment disorder and fluctuations, in particular at RT close to Tc. However, magnetic domains (~ 0.65 × 1.5 µm2) as depicted in Fig. 1d can still be observed in the film at that temperature by magnetic force microscopy44.

When the temperature increases towards Tc, the fluctuation of localized moments increases and the distribution of the µI orientations increases: the size of the magnetic domains decreases and the density of magnetic domain walls increases. The magnetization of the film decreases at the same time, in a similar way as observed in Fig. 3a. S decreases linearly during this process (Fig. 3b). Constant parameters A and B were determined experimentally from the measurements presented in Fig. 3b, considering47

$$S=-\frac{1}{eT}\left(A+B{k}_{B}T\right).$$
(4)

They are reported in Table 1. Assuming that ΔEC is negligible for Mn5Ge331,34 and combining Eqs. (3) and (4) one obtains A = βαΔEeI/2 and B = βΔR/2. Consequently, the linear behavior of S versus T suggests that the localized moment fluctuation effect occurs at constant β, constant αΔEeI, and constant ΔR, with βαΔEeI = 0.01 eV and βΔR ~  − 1/3. β depends on the DOS of spin-up and spin-down electrons (Fig. 1c) and should a priori vary with temperature. However, the energy variation related to a temperature change from 200 to 300 K is only ~ 8.62 × 10−3 eV, which may explain that the parameter β is found to be almost constant in our temperature range of investigation. ΔEeI is related to the energy gain involved with the polarization of free electrons by the magnetic moments localized on Mn ions. Considering that electron spin-up se+ =  + ½ and electron spin-down se− =  − ½, ΔEeI = ½ (JeI+ + JeI−)µI cosθ, with JeIj the exchange parameter of spin-up electrons (j = ‘ + ’) or of spin-down electrons (j = ‘ − ’), and θ the angle between the localized moment and the electron spin. JeIj and µI are expected to be independent of T in this model, and θ can be fixed to θ = 0 since ΔEeI corresponds to an average energy. Thus, ΔEeI should indeed be independent of T. α is also found to be independent of µI fluctuations with T close to Tc in Mn5Ge3. ΔR is related to the spin-dependence of free carrier scattering mechanisms. Indeed, the free carrier relaxation time τ is expected to be spin-dependent in FM materials, which is responsible for free carrier polarization P = (γ − 1)/(γ + 1) with γ = τ+/τ−. In our case, both Mn ions50 and magnetic domain walls can act as spin-dependent scattering centers51,52.

Table 1 Parameters A and B in Eq. (4) determined experimentally from S measurements presented in Fig. 3b for Mn5Ge3 films and Fig. 3e for MnxCoyGe1−x−y films.
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Thermal fluctuations of localized magnetic moments lead to the decrease of the spin-dependent S along with magnetization (and polarization). The increase of the product βαΔEeI promotes the increase of S, while the increase of the product βΔR leads to a decrease of S, since A > 0 and B < 0 (Table 1).

Magnetic moment ordering

Mn5Ge3 spin-dependent S variations versus temperature were also studied under an external magnetic field H applied in the film plane as shown in Fig. 1a. Figure 3b presents S variations versus temperature for four different magnetic field intensities H// = 8.6 × 10−3, 27 × 10−3, 46 × 10−3, and 73 × 10−3 Tesla in the direction parallel to the temperature gradient (Fig. 1a), as well as S measurements performed under an external magnetic field either oriented at 45° (H445° = 73 × 10−3 T), 90° (H5⊥ = 110 × 10−3 T), 135° (H4135° = 73 × 10−3 T), and 180° (H3180° = 46 × 10−3 T) compared to the temperature gradient direction (Fig. 1a). S variations versus temperature in presence of H are again found to be linear up to the FM/PA transition. The external magnetic field H promotes the increase of S at constant T, but the in-plane field effect is independent of the field direction and of the field intensity. Figure 3c shows the magnetization variations of the Mn5Ge3 film versus in-plane magnetic field intensity measured at 270 K (blue dashed line) and 200 K (solid squares). According to the magnetic hysteresis loop, the film magnetization was at saturation for all the S measurements performed under external magnetic field (H = HSAT), which can explain H effect on S to be independent of H intensity and orientation, considering the influence of the external field to be related to localized moment ordering and magnetic domain wall density variations (Fig. 1d). Maximum polarization of conduction electrons being reached at magnetization saturation, one should consider α = 1 in this case, and α < 1 otherwise. Considering α = 1 for S measurements performed on the Mn5Ge3 films under external magnetic field (Fig. 3b and Table 1), we obtain α = ½ without magnetic field, which is coherent with the considered model. The product βΔEeI = 0.02 eV at magnetization saturation. For comparison, the coupling energy of the localized moments should be of the order of kBTc = 0.026 eV (Fig. 3a). Comparing the values of B with and without external magnetic field (Table 1), the scattering parameter ΔRSAT (H = HSAT) is found to be twice as ΔR0 (H = 0), with ΔR0/ΔRSAT ~ ½ < 1. This means that the relaxation time difference Δτ between minority- and majority-spin electrons increases under magnetic field, and is found to scale with the parameter α in Mn5Ge3. These results agree with a higher polarization of conduction electrons at magnetization saturation. The polarization of free carriers has two opposite effects on the spin-dependent S: localized moment ordering promotes (i) an increase of A = βαΔEeI/2 due to a statistical increase of se and µI pairing, increasing S, and (ii) an increase of Δτ, decreasing S. However, the global effect of localized magnetic moment ordering in FM Mn5Ge3 promotes the increase of S at given T. H effect agrees with thermal fluctuation effect, as in the two cases, the spin-dependent S increases with localized moment polarization. This behavior can be the signature of bipolar conduction, since in the case of a FM film containing a single moment µI, M = (n+ − n−)µI with n+ and n− the concentrations of occupied states in each level (up and down).

Magnetic moment density

Figure 2a presents three diffractograms (b), (c), and (d), acquired on three MnCoGe films of different Mn composition. The films contain only the hexagonal MnCoGe phase (Supplementary Fig. S1b). They are polycrystalline and composed of columnar grains with a thickness ~ 152 ± 5 nm. The RMS surface roughness is ~ 0.9 nm and the MnCoGe grains exhibit an average lateral size ~ 1.8 × 0.7 µm2 for the three films MnCoGe, Mn0.34Co0.32Ge0.33, and Mn0.36Co0.29Ge0.34 according to AFM measurements (Fig. 2b). Figure 3d shows the magnetization variations versus temperature of the three MnxCoyGe1−x−y films. Magnetization and Curie temperature increase with x, which correspond to an increase of the density of localized moment µI and to an increase of the exchange energy between µI, respectively. Figure 3e presents the variations versus temperature of the spin-dependent S of the same films. Similar to Mn5Ge3, S is negative (n-type) and decreases linearly as temperature increases up to Tc for the three films. The FM/PA transition is more easily detected in the case of MnCoGe films, especially for Mn0.36Co0.29Ge0.34, S increasing with temperature after Tc. This change of behavior of S versus T is commonly observed in FM metals, S variations resulting from the bipolar effect of spin-up and spin-down electrons before Tc and S following the behavior of common metals after Tc 53. Furthermore, S is found to decrease as the Mn concentration increases in the MnCoGe films. The parameters A and B in Eq. (4) were determined from S measurements presented in Fig. 3e. They are displayed in Table 1. The slope of the linear function S = f(T) is independent of the Mn concentration with A = 0.001 eV for the three samples (Table 1), giving βαΔEeI = 0.002 eV if ΔEC = 0 in Eq. (3). This result suggests that the 3 at% increase of Mn atoms in the compound MnCoGe does not involve a significant modification of the MnCoGe DOS close to EF in our temperature range, supporting a constant parameter β, independent of Mn concentration. Furthermore, constant αΔEeI suggests also that the increase of exchange energy between localized moments µI has no influence on the average coupling energy of the conduction electron spin se with µI. Figure 3f presents S variations versus the Mn composition of MnxCoyGe1−x−y films. S is inversely proportional to x. This behavior is the result of the linear decrease of the parameter B as x increases in MnxCoyGe1−x−y films (Fig. 4a), suggesting an increase of Δτ with Mn concentration (Tab. 1), which is in agreement with the electrical conductivity behavior of MnxCoyGe1−x−y films versus x. Indeed, Fig. 4b shows that σ decreases as x increases. σ = enµe can decrease either due to a decrease of carrier density n, or due to a decrease of carrier mobility µe.

Figure 4
figure 4

Electrical properties of the MnxCoyGe1−x−y thin films. (a) electron-scattering-dependent parameter B (Eq. 4) as a function of x. The inset shows the kBTc energy variation versus x. (b) Electrical conductivity as a function of x, measured at different temperatures.

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However, β being independent of Mn concentration, the linear decrease of S versus x appears to be related to a decrease of µe = eτ/m*, due to a modification of the average scattering time τ in the two spin-dependent channels, resulting from a scattering effect associated with the additional Mn atoms. The increase of the magnetic moment density in MnCoGe have negligible effect on the exchange energy between conduction electron spin and localized µI, but promotes an increase of Δτ, leading to the decrease of S.

Extra Mn ions can act as additional spin-dependent scattering centers50, supporting a stronger spin-dependent scattering, which should be accompanied with a higher polarization of free electrons50. However, the polarization increase is not observed on the slope of the linear function S = f(T), such as in the case of Mn5Ge3 films under external magnetic field (Fig. 3b). Δτ could also be related to conduction electron bipolarity. Indeed, in the case of bipolar conduction, majority-spin electrons should diffuse towards the cold side (n-type S), while minority-spin electrons should diffuse towards the hot side, which could explain a difference of scattering conditions in the two channels independent of α. The decrease of S with the increase of Mn concentration should not be related to the corresponding polarization increase (Fig. 3d), since the spin-dependent S is observed to increase with magnetic polarization in both Mn5Ge3 (Fig. 3b) and MnxCoyGe1−x−y (Fig. 3e). Instead, the decrease of S with the increase of x is related to the corresponding increase of the direct exchange energy Eex =  − Jex µI.µI between Mn magnetic moments. Since µI coupling vanishes at T = Tc, Eex ~ kBTc can be assumed. The insert in Fig. 4a presents the variations of the product kBTc versus x. Eex is found to increase linearly with the Mn content. Consequently, free electron scattering in FM MnCoGe mainly involves coupled Mn magnetic moments, explaining the linear increase of the spin-dependent scattering difference (ΔR in Eq. 3) between the two spin-dependent channels with Mn composition by the linear increase of moments’ exchange energy.

Outlook

The influence of thermal fluctuations, ordering, and density of localized magnetic moments on the spin-dependent Seebeck coefficient S has been investigated in ferromagnetic Mn5Ge3 and MnCoGe thin films. S is found to be mainly sensitive to the polarization of the conduction electrons, according to two opposite mechanisms: (i) the coupling of the spin vector of free carriers with the localized moments leads to higher S, while (ii) an increase of the scattering difference in the two spin-dependent conduction channels leads to lower S. In the case of Mn5Ge3 films under external magnetic field, the overall contribution of the two mechanisms leads to an increase of S. The thermal fluctuations of localized magnetic moments, as well as moment ordering using an external magnetic field show coherent effects, as in both cases S increases with localized moment polarization. The increase of Mn concentration in hexagonal MnCoGe leads to an increase of the magnetic moment density and of the localized moment exchange energy. The moment density increase is found to have no effect on the average exchange energy between the spin of conduction electrons and localized moments, but promotes an increase of the relaxation time difference between the two spin-dependent conduction channels, leading to the linear decrease of S as the Mn content increases. Our observations in model materials coupled with the expression proposed in Eq. (3) open prospects for original engineering routes for the development of spin-engineered thermoelectric materials.

Materials and methods

Mn5Ge3 and MnCoGe films were elaborated by magnetron sputtering and solid-state reaction. 99.99% pure Co, 99.99% pure Ge, and 99.9% pure Mn targets were sputtered on 1.5 × 2.5 cm2 glass substrates in a commercial magnetron sputtering system with a base vacuum of 10−8 Torr28. In this system, three targets placed at an angle of 45° to the normal of the sample surface can be simultaneously sputtered during the substrate rotation. Co, Ge, and Mn deposition rates were separately calibrated thanks to the measurement by X-ray reflectivity (XRR) of the thickness of different films deposited in different conditions. The substrates were cleaned 10 min in an acetone bath before to be rinsed 10 min in alcohol in an ultrasonic cleaner. They were finally kept 30 min at 423 K in a baking furnace, before to be loaded in the sputtering chamber. The elements were deposited (Ge and Mn for Mn5Ge3) or co-deposited (Mn, Co, and Ge for MnCoGe) at room temperature on the glass substrates that were rotated at 5 rpm. The Mn5Ge3 films were produced by reactive diffusion: 35 nm of Ge were deposited on the glass substrate before to be capped with 31 nm of Mn30. After deposition, the samples were ex situ annealed under vacuum (P ~ 10−7 mbar) at 400 °C for 10 min. The diffractogram (a) in Fig. 2a shows some diffraction peaks belonging to the Ge lattice in addition to the peaks of the phase Mn5Ge3, meaning that the Ge layer was not entirely consumed by the growth of Mn5Ge3 despite the entire consumption of the Mn layer. The MnCoGe films were produced by non-diffusive reaction28, allowing the stoichiometry of the compound to be varied: Mn, Co, and Ge were co-sputtered on the substrate up to a thickness of 150 nm. The samples were also ex situ annealed at 400 °C for 10 min after deposition.

Sample structure was investigated by both X-ray diffraction (XRD) in the Bragg–Brentano geometry (θ − 2θ) using a Cu Kα source (λKα = 0.154 nm) in a PANalytical X’Pert PRO setup equipped with an X’Celerator detector, and by atomic force microscopy (AFM) using a Solver-PRO system from NT-MDT. Film magnetization was measured versus temperature using a SQUID magnetometer Quantum Design MPMSXL. Hall measurements and sample resistivity were measured in the Van der Pauw geometry using a lab-made setup operating between 20 and 350 K. The applied magnetic field for Hall measurements was 0.5 T. The Seebeck coefficients of the films were measured using a home-made setup39 between T = 225 and 325 K, close to the FM/PM transition, according to the geometry presented in Fig. 1a. The distance d between the two electrodes was 1 cm.