Coherent Ag-rich nanoprecipitates/β-Ag₂Se flexible film with unprecedented thermoelectric performance by liquid-like sintering

Abstract

β-Ag₂Se holds a great promise for wearable thermoelectric generators due to its good mechanical properties and biocompatibility. However, optimizing its power factor through alterations in the Ag/Se stoichiometric ratio or incorporating secondary phases has reached the limitation. This work adopts a liquid-like sintering strategy, namely employing the migration and precipitation of silver ions from metallic silver to the silver selenide during the spark plasma sintering process to construct significantly sized and coherent Ag-rich nanoprecipitates within flexible Ag/β-Ag₂Se composite films, thereby achieving an unprecedented power factor exceeding 4000 μWm⁻¹K⁻² at 303 K. The Ag-rich nanoprecipitates play a crucial role in elevating the carrier concentration, enhancing the density-of-states effective mass, and mitigating carrier scattering caused by phase interfaces and acoustic phonons. The generator containing 5 pieces of optimal films shows an normalized output power density of 10.89 μWcm⁻¹K⁻² under a temperature difference of 26.0 K. This study strongly suggests that the creation of Ag-rich nanoprecipitates is an effective avenue for improving the electrical performance of β-Ag₂Se-based thermoelectric materials.

Introduction

The rapid progress in flexible electronics has catalyzed an exigent need for power sources adaptable to the inherent flexibility of these systems¹. Flexible thermoelectric films, capable of conforming to the skin and continuously converting temperature difference between the human body and the ambient environment into electrical energy for flexible electronics, are viewed as a highly promising flexible self-powered energy source^2,3. Optimizing the power factor (PF) of the flexible thermoelectric film is the key to improving its output power density to meet the energy supply needs of as many electronic products as possible⁴.

Due to its better mechanical properties and lower toxicity compared to the traditional BiTeSe alloys, N-type silver selenide (β-Ag₂Se at room temperature) has attracted sharply increasing attention in the field of flexible thermoelectric films^5,6. However, although its Hall carrier mobility (μ_H) reaching up to 2000 cm²V⁻¹s⁻¹ at room temperature, the values of density-of-states effective mass (m_d) for carriers in the stoichiometric β-Ag₂Se are far smaller than that of BiTeSe alloys^7,8,9. Furthermore, owing to the doping limits of most elements (including Ag and Se) in β-Ag₂Se being less than 0.1 mole%, the intrinsic small carrier concentration (n_H) of stoichiometric β-Ag₂Se can hardly be effectively elevated through doping or altering the Ag/Se ratio since the 1960s^10,11,12,13. Thus, the state-of-art values of PF for stoichiometric β-Ag₂Se, in both bulk and film forms, typically remain below 2700 μWm⁻¹K⁻² or even 2000 μWm⁻¹K⁻² at around room temperature for decades, which is markedly inferior to the exceeding 4000 μWm⁻¹K⁻² obtained by BiTeSe alloys^4,14,15,16. On the other hand, despite incorporating metals or alloys as the second phase has been proven to enable the promotion of n_H and enhancement of m_d in β-Ag₂Se-based flexible films^17,18, the values of PF for these films are still limited by the excessively high n_H and additional carrier scattering at phase boundaries^19,20,21,22.

Very recently, we revealed that a previously overlooked microstructure, Ag-rich nanoprecipitates with a size of a few nanometers can be found in nominally stoichiometric β-Ag₂Se sintered by spark plasma sintering (SPS)²³. However, possibly because of their limited size and content, the nanoprecipitates did not exert a substantial influence on the thermoelectric properties of β-Ag₂Se. In this work, addressing the mobility of Ag ions and its low solubility in β-Ag₂Se, a liquid-like SPS approach is proposed, wherein Ag ions migrate from metallic silver to silver selenide during the sintering process, subsequently bonding partially with selenium atoms and precipitating out to form Ag-rich nanoprecipitates with sizes reaching several tens of nanometers (marked with red arrows in Fig. 1d) in flexible Ag/β-Ag₂Se film (see “Methods”, Supplementary Fig. 1 and Fig. 1a–d). As shown in Fig. 1e, the boundary between the Ag-rich nanoprecipitate and β-Ag₂Se matrix is completely coherent due to their identical crystal structure of orthorhombic Ag₂Se. The electrons transferred from the Ag-rich nanoprecipitates and metallic Ag to the β-Ag₂Se matrix collectively pushed up the n_H of composite films (see Supplementary Fig. 2), and the Ag-rich nanoprecipitates also play a significant role in enhancing the m_d and diminishing carrier scattering caused by phase interfaces and acoustic phonons. Thus, as depicted in Fig. 1f, despite a high n_H of 22.16 × 10¹⁸ cm⁻³, this optimal film (named as Ag_2.3Se-SPS200 °C/5 min) maintains a moderate μ_H of 855.4 cm²V⁻¹s⁻¹. Meanwhile, because of the appropriate n_H and m_d, the optimal film retains a Seebeck coefficient of −116.3 μVK⁻¹, showing only a marginal decline compared to that of pristine β-Ag₂Se (see Fig. 1g). As a result, via this novel strategy that can transform metallic Ag into the Ag-rich nanoprecipitates, an unprecedented PF exceeding 4000 μWm⁻¹K⁻² has been achieved in the optimal film, setting a new benchmark for the electrical performance of β-Ag₂Se based thermoelectric materials (see Fig. 1h). In addition, regardless of whether the length of the film is taken into account, the normalized output power density (P_md) of the thermoelectric generator formed by 5 pieces of optimal films (1 cm in length) significantly surpassing that of other β-Ag₂Se based film-type thermoelectric generators (see Fig. 1i).

Fig. 1: Formation of Ag-rich nanoprecipitates and electrical properties of optimal composite film in this work.

a Schematic diagram of the fabrication of nylon-supported flexible composite films. b–d EDS maps of Ag element (b) and Se element (c), and bright-field TEM image (d) for Ag_2.3Se-SPS200 °C/5 min film (typical Ag-rich nanoprecipitates denoted with red arrows). e High-resolution transmission electron microscopy (HRTEM) image showing the coherent boundary (marked with red dotted circle) between the Ag-rich nanoprecipitate boundary and the β-Ag₂Se matrix. All the lattice spaces marked with the white lines correspond to the planes of orthorhombic β-Ag₂Se. f–h Comparison of electrical properties between the optimal film in this work and other β-Ag₂Se based bulks and films. (1 ~ 10) Different β-Ag₂Se based bulks and films include 1: Ag/Ag₂Se film²⁰, 2: Ag/Ag₂Se film²¹, 3: Ag₂Se film³², 4: Ag₂Se bulk⁷, 5: Ag_2.05Se film²², 6: Ag₂Se_1.01 bulk³³, 7: Ag₂Se_1.06 bulk¹³, 8: (Ag_0.998Cu_0.02)₂Se bulk¹⁴, 9: Ag₂Se bulk³⁴, 10: Ag_1.8Se film¹⁵. i Comparison of normalized output power density between the thermoelectric generator in this work and other β-Ag₂Se-based film-type thermoelectric generators^{15,17,18,19,21,26,32,35,36}.

Results and discussion

By systematically investigating the composition and microstructure of composite films fabricated from composite powders of varying compositions and subjected to different sintering processes, the potential formation mechanism of Ag-rich nanoprecipitates can be deduced. According to the molar ratio of AgNO₃ to Se powder in the solvothermal reaction, and the temperature and duration of SPS, the obtained thermoelectric films are named as Ag_2.0Se-SPS200 °C/5 min, Ag_2.1Se-SPS200 °C/5 min, Ag_2.3Se-SPS140 °C/1 min, Ag_2.3Se-SPS200 °C/5 min (the optimal film), Ag_2.3Se-SPS200 °C/30 min, and Ag_2.5Se-SPS200 °C/5 min.

Since the Ag-rich nanoprecipitates with notable size in this work have never been reported in the stoichiometric β-Ag₂Se, it is speculated that the presence of sufficient metallic Ag in the composite powder before sintering is a prerequisite for the formation of Ag-rich nanoprecipitates. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images in Supplementary Fig. 3 confirm this inference. In HAADF-STEM image of Ag_2.1Se-SPS200 °C/5 min, hardly any notable Ag-rich nanoprecipitates can be observed, while HAADF-STEM images of Ag_2.3Se-SPS200 °C/5 min and Ag_2.5Se-SPS200 °C/5 min distinctly reveal the Ag-rich nanoprecipitates.

Composition and microstructure of films fabricated using Ag_2.3Se powder have been investigated to illustrate the formation process of Ag-rich nanoprecipitates. For β-Ag₂Se particles in the Ag_2.3Se powder before sintering, the atomic fraction of Ag element is determined to be 67.69% by energy dispersive X-ray spectroscopy (EDS) maps (see Supplementary Fig. 4). As depicted in the dark-field TEM image presented in Fig. 2a, for Ag_2.3Se-SPS140 °C/1 min film, numerous circular “bright” domains (typically marked with white arrows), primarily within 10 nm in size can be recognized by the strain field contrast. The result of EDS line scanning across a “bright” domain (see Supplementary Fig. 5) in Fig. 2d reveals that the average atomic fraction of Ag element in the “bright” domain (75.08%) is higher than that in the surrounding β-Ag₂Se matrix (71.02%), namely, these domains are Ag-rich. Notably, the β-Ag₂Se matrix also exhibits an elevated average atomic fraction of Ag element than that of the β-Ag₂Se particles in the Ag_2.3Se powder. For the composite films sintered at 200 °C for 5 min and 30 min, as shown in Fig. 2b, c, a large number of island-like nanoprecipitates spanning sizes from tens to approximately 100 nanometers can be observed. Determined by EDS maps of the typical nanoprecipitates (marked with red dashed lines) and β-Ag₂Se matrix (marked with yellow dashed lines), for Ag_2.3Se-SPS200 °C/5 min film, the average atomic fractions of Ag element in the nanoprecipitates and β-Ag₂Se matrix are 69.42% and 67.15%, respectively. For clear comparison, the average atomic fractions of Ag element in the nanoprecipitates and β-Ag₂Se matrix for all samples in this work are listed in Supplementary Table 1. As to Ag_2.3Se-SPS200 °C/30 min film, the corresponding values increase to 73.49% and 68.81%, respectively. HRTEM images in Fig. 2e, f demonstrate that the boundaries between the Ag-rich nanoprecipitates and the surrounding matrix (indicated by red dashed lines) are coherent, and the Ag-rich nanoprecipitates exhibit pronounced Moiré fringes with varying spacings. The fast Fourier transform (FFT) patterns inserted in Fig. 2a, e, f, while displaying clear secondary diffraction features, can all be indexed to the orthorhombic phase of Ag₂Se.

Fig. 2: Microstructure and composition of Ag-rich nanoprecipitates.

a Dark-field HRTEM image of Ag_2.3Se-SPS140 °C/1 min film inserted with a FFT pattern. b, c HAADF-STEM images Ag_2.3Se-SPS200 °C/5 min film and Ag_2.3Se-SPS200 °C/30 min film. d Atomic fraction of Ag element revealed by EDS elemental line scan across a “bright” domain. e, f Dark-field HRTEM image of Ag_2.3Se-SPS200 °C/5 min film and Ag_2.3Se-SPS200 °C/30 min film inserted with the FFT patterns.

Rietveld refined X-ray diffraction patterns have been employed to estimate the mass fraction of metallic Ag in the Ag_2.3Se composite powder and the sintered films (see Supplementary Fig. 6 and Supplementary Table 2). All the diffraction peaks can be indexed to orthorhombic β-Ag₂Se (JCPDF card number: 01-071-2410) and cubic metallic Ag (JCPDF card number: 03-065-2871), and Ag_2.3Se composite powder contains approximately 8 wt% of metallic Ag. Nevertheless, as to the film sintered at 140 °C (slightly higher than the phase transition temperature (133 °C) of β-Ag₂Se) for only 1 min, the Ag and Ag₂Se particles quickly fused with each other to form bulks (see Supplementary Fig. 7), and the mass fraction of metallic Ag drops to about 4%. Further increasing the sintering temperature to 200 °C and extending the sintering durations to 5 and 30 min facilitated complete sintering of the composite films, which led to a reduction in the thickness of composite films and a shrinkage in size of the metallic Ag regions within the composite films (see Supplementary Figs. 8 and 9), but the mass fractions of metallic Ag in these films still remain around 4%. Hence, it is inferred that the excess Ag atoms in the Ag-rich nanoprecipitates stem from the metallic Ag, and the transfer of Ag atoms from metallic Ag to Ag₂Se is completed during the phase transition.

Drawing from the above discussion, the presumed process underlying the formation of the Ag-rich nanoprecipitates during sintering is illustrated in Fig. 3a. In comparison to the Ag_2.3Se powder, the mass fraction of metallic Ag in the Ag_2.3Se-SPS140 °C/1 min film is diminished. Conversely, the atomic fraction of Ag element within the β-Ag₂Se matrix is promoted, accompanied by the formation of Ag-rich domains. Therefore, during the phase transition of β-Ag₂Se, part of Ag atoms migrates from the metallic Ag into the interstitial sites of Ag₂Se, forming the nanosized Ag-rich domains in the β-Ag₂Se matrix. This state is inherently unstable due to the extremely low solubility of silver atoms in Ag₂Se. Consequently, the excess Ag atoms at interstitial sites tend to precipitate out, and the Ag-rich domains represent the early nucleation stage of the nanoprecipitates. As the sintering temperature and duration rise to 200 °C and 5 min, respectively, Ag atoms continuously precipitate out from the interstitial sites in Ag₂Se and bond with Se atoms to form Ag-rich nanoprecipitates. This supposition accounts for the observed reduction in the atomic fractions of Ag elements within the β-Ag₂Se matrix and Ag-rich nanoprecipitates when compared to the film sintered at 140 °C for 1 min. Meanwhile, the feasibility of this supposition has also been confirmed through theoretical models (see Fig. 3b). The model of perfect (11-2) plane of β-Ag₂Se containing 72 atoms (Ag₄₈Se₂₄) was firstly created, then, 7 Ag atoms were inserted into the interstitial sites of this model (marked with the red dashed oval). After the relaxation of the model inserted with excess Ag atoms (Ag₅₅Se₂₄), the inserted Ag atoms moved from the interstitial sites to the surface of β-Ag₂Se to form the Ag-rich nanoprecipitate (marked with the green dashed oval), which is coherent with the (11-2) plane of β-Ag₂Se. Based on the models, the average values of Bader charge for Ag atoms in metallic Ag, β-Ag₂Se, and Ag-rich nanoprecipitate are calculated to be 11.001, 10.772, and 10.844, respectively, suggesting that the Ag atoms in Ag-rich nanoprecipitate are partially bonded with the Se atoms. On the other hand, in Fig. 3a, with the extension of sintering duration to 30 min, the mass fraction of metallic Ag in the composite film remains at around 4 wt%, implying that the observed elevation in the atomic fraction of Ag element within the β-Ag₂Se matrix and Ag-nanoprecipitates is not attributable to metallic Ag, but rather to Se loss resulting from volatilization. The volatilization of Se elements leads to a sharp increase in n_H (50.4 × 10¹⁸cm⁻³), causing a dramatic decline in the Seebeck coefficient and PF of the Ag_2.3Se-SPS200 °C/30 min to −58.33 μVK⁻¹ and 2338.96 μWm⁻¹K⁻², respectively. Thus, precise control of the SPS process plays a decisive role in regulating thermoelectric performance of Ag₂Se-based materials.

Fig. 3: The formation mechanism of Ag-rich nanoprecipitates during the SPS process.

a Schematic diagram revealing the possible microscopic reactions in the composite film at different stages of SPS process. b Theoretical models to verify the possible formation mechanism of Ag-rich nanoprecipitates. Blue balls represent Ag atoms and yellow balls represent Se atoms.

To elucidate the factors contributing to the record-high PF observed in Ag_2.3Se-SPS200 °C/5 min film, the carrier transport characteristics of this film have been thoroughly analyzed and compared with that of other β-Ag₂Se-based materials. Firstly, the higher n_H observed in Ag_2.3Se-SPS200 °C/5 min film compared to single-phase β-Ag₂Se-based materials at or near stoichiometry is reasonable. As per the theoretical models outlined in Fig. 3b, the Ag-rich nanoprecipitates can be fundamentally conceptualized as β-Ag₂Se with Se vacancies, naturally exhibiting a higher concentration of electron compared to stoichiometric β-Ag₂Se. Moreover, the computed charge density differences suggest that both the Ag-rich nanoprecipitate and metallic Ag are capable of electron donation to β-Ag₂Se, hence, enhancing the abundance of Ag-rich nanoprecipitates and metallic Ag within the composite films in this work can bolster the n_H. However, an excessive presence of Ag-rich nanoprecipitates alongside metallic Ag would lead to a remarkably heightened n_H and a notable decline in the PF (see Supplementary Note 1 and Supplementary Tables 1–5). Secondly, in order to figure out why Ag_2.3Se-SPS200 °C/5 min maintains a high n_H while preserving a μ_H close to 1000 cm²V⁻¹s⁻¹, the values of activation energy (E_b) characterizing the barrier height of grain boundaries and phase interfaces in the β-Ag₂Se-based materials were estimated using the equation ${\mu }_{H}=\frac{{el}}{\sqrt{8{k}_{B}T\pi {m}_{d}}}\exp ({-E}_{b}/{k}_{B}T)$ (see Fig. 4a), where e is the elementary charge, l is the average grain size (set as 100 μm for the calculation), k_B is the Boltzmann constant and T is the absolute temperature (set as 300 K for the calculation)²⁴. Clearly, due to the existence of interfacial barrier in the composites, the composites show larger values of E_b than the single-phase β-Ag₂Se-based materials at or near stoichiometry. Of particular interest is the observation that the Ag_2.3Se-SPS200 °C/5 min film in this work exhibits E_b akin to those observed in single-phase β-Ag₂Se-based materials, unlike other composites. This suggests that the coherent grain boundaries between the Ag-rich nanoprecipitate and β-Ag₂Se have negligible effects on carrier scattering, and the carriers are also not significantly scattered by the interfaces between metallic Ag and β-Ag₂Se, because of the relatively small mass fraction (about 4%) of metallic Ag in the film. Thus, to circumvent excessive interfacial scattering, precise regulation of the quantity of secondary phases in β-Ag₂Se-based materials is crucial. Further, the values of deformation potential, indicative of the interaction strength between acoustic phonons and carriers, were evaluated for the Ag_2.3Se-SPS200 °C/5 min film in this work and other Ag₂Se-based materials with similar scattering mechanism using the single parabolic band (SPB) model. Based on the values of m_d and E_def, the n_H dependent theoretical μ_H and PF were calculated and plotted (see “Methods” and Fig. 4b, c). As shown in Fig. 4b, compared to single-phase β-Ag₂Se-based materials, Ag_2.3Se-SPS200 °C/5 min film despite having the maximum value of m_d (0.510 m₀), shows only two-third to half of the E_def for single-phase β-Ag₂Se-based materials, indicating minimal scattering of carriers by acoustic phonons in it. Thus, it can be summarized that even with the high n_H and large m_d, Ag_2.3Se-SPS200 °C/5 min film maintains a moderate μ_H for two reasons, namely the mitigation of carriers scattering at phase interfaces, and the reduction in scattering of carriers by acoustic phonons. The plots in Fig. 4c reveal that irrespective of adjustments to n_H, the maximum theoretical PF at 300 K of the reported single-phase β-Ag₂Se-based materials is limited at approximately 3000 μWm⁻¹K⁻², which is predominantly attributed to the relatively large E_def. On the contrast, leveraging the diminished E_def, the PF of Ag_2.3Se-SPS200 °C/5 min film holds promise to exceed 4500 μWm⁻¹K⁻² if further optimization of n_H is achieved. In fact, some of the representative Ag_2.3Se-SPS200 °C/5 min films and bulks in this work indeed have realized such a large PF (see Supplementary Fig. 10), and the overall relative standard deviation of the PF values for these samples is calculated to be 5.13% (see Supplementary Table 6), indicating good repeatability in performance. To illuminate the reasons behind the larger m_d and smaller E_def of the Ag_2.3Se-SPS200 °C/5 min film compared to single-phase β-Ag₂Se-based materials, based on the theoretical models (see Supplementary Fig. 11), the electronic band structure and theoretical E_def for stoichiometric β-Ag₂Se and Ag-rich nanoprecipitate have been obtained (see “Methods”, Fig. 4d, e, f). As revealed by Fig. 4d, e, the bottom of conduction band for β-Ag₂Se appears relatively sharp, and the corresponding band-edge effective mass (m_b) is 0.265 m₀, while the Ag-rich nanoprecipitate displays a markedly flattened bottom of conduction band, with the m_b of 0.414 m₀. Additionally, by calculating the strain-dependent variations of the conduction band minimum (CBM) in Fig. 4f, the values of theoretic E_def for stoichiometric β-Ag₂Se and Ag-rich nanoprecipitate are 11.49 eV and 5.15 eV, respectively. Therefore, the formation of Ag-rich nanoprecipitates in the composite films in this work leads to the larger m_d and smaller E_def.

Fig. 4: Carrier transport characteristics and formation of Ag-rich nanoprecipitates and electrical properties of optimal composite film in this work.

a Barrier height of grain boundaries and phase interfaces for the optimal film in this work and other β-Ag₂Se based bulks and films. (1–10) Different β-Ag₂Se based bulks and films include 1: (1–10) Different β-Ag₂Se based bulks and films include 1: Ag₂Se_1.01 bulk³³, 2: Ag₂Se bulk⁷, 3: (Ag_0.998 Cu_0.02)₂Se bulk¹⁴, 4: Ag₂Se bulk³⁴, 5: Ag₂Se_1.06 bulk¹³, 6: Ag₂Se film³², 7: Ag_1.8Se film¹⁵, 8: Ag/Ag₂Se film²¹, 9: Ag_2.05Se film²², 10: Ag/Ag₂Se film²⁰. b, c Calculated n_H dependent theoretical μ_H and PF plots for the optimal film in this work and other β-Ag₂Se based bulks and films with similar scattering mechanism. The values of m_d and E_def include 1: m_d = 0.211 m_e, E_def = 7.605 eV, 2: m_d = 0.186 m_e, E_def = 10.733 eV, 3: m_d = 0.221 m_e, E_def = 9.120 eV, 4: m_d = 0.219 m_e, E_def = 9.393 eV, 5: m_d = 0.298 m_e, E_def = 6.653 eV, 6: m_d = 0.276 m_e, E_def = 7.541 eV, 7: m_d = 0.248 m_e, E_def = 9.071 eV. d–f Calculated electronic band structures and strain-dependent variations in the conduction band minimum of β-Ag₂Se matrix and Ag-rich nanoprecipitate.

For the potential application of flexible thermoelectric films, besides the PF, the length of thermoelectric film should also be optimized to maximize the output power density. In this work, the length of a single Ag_2.3Se-SPS200 °C/5 min film with width of 5 mm and thickness of 30 μm is optimized by numerical simulation. The model for simulation is shown in Fig. 5a, and the parameters for simulation are listed in Supplementary Table 7. Additionally, the contact resistance (R_c) between the electrodes (silver paste) and Ag_2.3Se-SPS200 °C/5 min film is measured to be 3.92 mΩ (See Supplementary Fig. 12). The ambient temperature (T_a) is 298.15 K, and the heat exchange coefficient between the film and air is set as 4 W m⁻² K⁻¹. The length of the silver electrodes on both ends of the film is 2 mm. The thermal conductivity (κ) of Ag_2.3Se-SPS200 °C/5 min film (1.77 Wm⁻¹K⁻¹) is calculated as κ = ρDC_p, where ρ is the density (8198.4 kgm⁻³), D is the thermal diffusivity and C_p is the specific heat capacity (see Supplementary Fig. 13). Therefore, ZT of this film at 303 K is calculated to be 0.7, which is comparable to the commonly reported ZT of Ag₂Se based bulk materials. The electrical and thermal properties of nylon substrate and silver electrodes are obtained from the software (Comsol Multiphysics 5.6) for simulation and the thickness of nylon substrate is 35 μm. The temperature gradient and electrical potential field have been simulated for the film with temperature of hot side (T_h) set as 308.15 K and 318.15 K, respectively. Figure 5b shows the simulated results for the film with T_h set as 308.15 K. The temperature of the cold side (T_c) decreases from 306.45 K to 298.23 K as the length of the film increases from 1 mm to 18 mm, and the open circuit voltage (U_oc) increases from 0.19 mV to 1.14 mV. The maximum output power (P_m) is expressed as$\,{P}_{{\rm{m}}}=\,\frac{{U}_{{\rm{oc}}}^{2}}{4{\times }({R}_{{\rm{f}}}+{2\times} ({R}_{{\rm{c}}}+{R}_{{\rm{e}}}))}$, where R_f and R_e the theoretical resistance of the Ag_2.3Se-SPS200 °C/5 min film and silver paste. The simulated highest P_m of 1.40 μW is realized in the film with a length of 6 mm, and as displayed in Fig. 5c, the value of U_oc for this film is 8.86 × 10⁻⁴V. Likewise, in Fig. 5d, with the increasing length of film, the T_c drops from 314.76 K to 298.31 K for the T_h of 318.15 K. In addition, the simulated U_oc climbs from 0.39 mV to 2.28 mV and a peak value for P_m of 5.58 μW is realized in the film. In Fig. 5e, the corresponding value of U_oc is 1.77 × 10⁻³ V. Moreover, the theoretical values of energy conversion efficiency (η_T) of the single-film generator with T_h of 308.15 K and 318.15 K are 0.36% and 0.71%, respectively (see Supplementary Note 2 and Supplementary Fig. 14).

Fig. 5: Output performance of single optimal film and the thermoelectric generator composed of 5 pieces of optimal films.

a Model of single Ag_2.3Se-SPS200 °C/5 min film for numerical simulation. b–e Simulated results and potential distribution for the film with T_h set as 308.15 K (b and c) and 318.15 K (d and e). f, g Experimentally measured output voltage (f) and power (g) of the thermoelectric generator composed of 5 pieces of optimal films.

It is noteworthy that, although the natural convection condition at the cold end represents the most probable practical scenario for film-type wearable thermoelectric generators, currently, the output performance of flexible film-type thermoelectric generators are predominantly measured under conditions of fixed values of ΔT. In this work, in order to facilitate and ensure the utmost accuracy in the preparation and measurement of the film-type generator, as well as to enable an objective comparison with the output performance reported in other works, 5 pieces of Ag_2.3Se-SPS200 °C/5 min film with a length of 1 cm were connected in series to form a thermoelectric generator (see Supplementary Fig. 15), and the output performance of this generator was then tested with different values of ΔT. The output voltage and power of this generator are shown in Fig. 5f, g, respectively. Both the open-circuit voltage and maximum output power increase with the rise in the ΔT across the generator. For the ΔT of 26.0 K and 57.0 K, the measured P_m attains 55.23 μW and 204.78 μW, respectively. In addition, depending on whether the length of the film (L) is taken into account, there are two methods for the calculation of normalized output power density, namely ${P}_{{\rm{md}}}=\,\frac{{P}_{{\rm{m}}}}{{A}_{{\rm{TEG}}}{\Delta T}^{2}}$^18,25, or ${P}_{{\rm{md}}}=\,\frac{{P}_{{\rm{m}}}L}{{A}_{{\rm{TEG}}}{\Delta T}^{2}}$^26,27, where A_TEG is the cross-sectional area of the thermoelectric generator. According to these two expressions, the highest P_md this work is calculated of 10.89 (μWcm⁻²K⁻² or μWcm⁻¹K⁻²) with ΔT = 26.0 K.

The mechanical properties of the Ag_2.3Se-SPS200 °C/5 min materials have also been investigated. After 800 bending cycles (the radius for bending is 2.5 mm), an increase in resistivity of the bent film (ρ_b) is observed, amounting to approximately a 10% elevation compared to its pristine resistivity (ρ₀), and then keep almost constant during the subsequent bending cycles. On the other hand, throughout the entire flexibility test, the variation in the Seebeck coefficient of the bent film (S_b) remained within ±4% of its original value (S₀), suggesting that the bending cycles can hardly affect the Seebeck coefficient of the film (see Supplementary Fig. 16a). Moreover, the bending strength of Ag_2.3Se-SPS200 °C/5min-based bulk is measured to be 128.69 MPa (see Supplementary Fig. 16b), which is very close to that of β-Ag₂Se reported by Pei’s group and significantly exceeds that of (Ag_0.998Cu_0.02)₂Se composite and Ag₂Se/CNTs composite^6,14,28.

Via a liquid-like SPS process, Ag-rich nanoprecipitates with notable sizes and proper composition have been created in the flexible Ag/β-Ag₂Se composite films. The formation of the Ag-rich nanoprecipitates is speculated to arise from the migration of Ag atoms from metallic Ag to the interstitial sites of the Ag₂Se lattice, where they bond with Se atoms and precipitate out. Ag-rich nanoprecipitates can positively influence the thermoelectric performance of composite films by not only increasing the n_H and improving the m_d, but also reducing carrier scattering from phonons and phase boundaries. An unprecedented and stable power factor exceeding 4000 μWm⁻¹K⁻² at room temperature has been realized in the optimal film. Moreover, the thermoelectric generator composed of 5 pieces of optimal films achieved an normalized output power density up to 10.89 μWcm⁻¹K⁻². This work has presented an innovative research direction for further enhancing the thermoelectric performance of β-Ag₂Se-based materials.

Methods

Materials

Silver nitrate (AgNO₃, ≥99.8%) was purchased from XiLong Scientific Co., Ltd. Selenium powder (Se, ≥99.9%) and ethylenediamine (C₂H₈N₂, 99%) were purchased from Aladdin Chemical Co., Ltd. Ethylene glycol and anhydrous ethanol were bought from Sinopharm Chemical Reagent Co., Ltd. The glass-fiber sheets were purchased from Yuyan (Shanghai) Chemical Co. and the nylon membrane was produced by Yibo (Zhejiang) Filtration Equipment Co., Ltd. All these materials were used without further purification.

Synthesis of Ag/β-Ag₂Se powders

At room temperature, a certain amount of Se powder and AgNO₃ were dissolved in 12.5 mL and 5 mL of ethylenediamine, respectively. Then these two solutions were mixed and maintained at 180 °C for 5 h in a sealed 25 mL Teflon-lined stainless-steel autoclave. After the autoclave was naturally cooled to room temperature, the obtained dark grey powder was collected by centrifugation and washed with ethanol and deionized water for three times. The total amount of the AgNO₃ and Se powder for synthesis of all samples is 0.4187 g and molar ratio of AgNO₃ and Se varies from 2.0 to 2.5 to adjust the mass fraction of metallic Ag in the resulted composite powders.

Fabrication of composite films and bulks

The resulted powder was uniformly dispersed in 10 mL ethylene glycol and 1.2 mL of the dispersion was drop-casted on a piece of glass-fiber sheet with size of 5 mm × 20 mm. After the glass-fiber sheet coated with composite powder was dried in vacuum at 80 °C for 10 h, it was sandwiched between 2 pieces of nylon membrane and sintered at 30 MPa. During the sintering process, the composite powder was insulated from the pulse current of SPS equipment by the nylon membrane, avoiding the migration of Ag ions caused by the pulse current. The powder was transformed into the nylon-supported dense composite film during the sintering process, and the debris of glass-fiber sheet could be easily removed with a brush. To demonstrate that the nylon substrate has no influence on the thermoelectric properties of the films, some composite bulk materials were also prepared by sintered the composite powder in the conditions same with the fabrication of films, but without using the glass-fiber membrane and nylon film.

Material characterizations and property measurements

The phase and composition of composite powders and films were analyzed by X-ray diffractometer using Cu Ka radiation (Rigaku, SmartLab). The mass fractions of Ag element and Se element in samples were acquired by inductively coupled plasma optical emission spectrometry (ICP-OES, Agilent 720). The morphology and thickness of composite powders and films were revealed via field emission scanning electron microscopy (Hitachi S-4800). The microstructures were observed by transmission electron microscopy (FEI TalosF200X), with an Oxford energy dispersive X-ray spectrometer (EDS) to analyze the elemental compositions and distributions of the samples. The simultaneous measurement of the Seebeck coefficient and electrical conductivity at room temperature were carried out in an Ulvac-Riko ZEM-3 system. The measurement was conducted in argon atmosphere with temperature gradients of 20, 30, and 40 °C. The carrier mobility and carrier concentration were collected using a Hall effect system (Lake Shore Cryotronics 8404). A laser flash diffusivity apparatus (Netzsch LFA 467) and a differential scanning calorimeter (Netzsch 404 F3) were applied to determine the thermal diffusivity and specific heat capacity, respectively. The density was measured via the Archimedes’ method. Since the measured thermal diffusivity of composite film is dramatically lower than the reasonable value, pellets were sintered using Ag/β-Ag₂Se powder in the conditions same with the fabrication of films for the measurement of thermal diffusivity and density. Each pellet is circular, with a thickness of 0.5 mm and a diameter of 12.7 mm. The sound velocity was measured via a sound velocimeter (JSR, DPR300). The hot and cold side temperatures of single composite film were measured employing a multiplex temperature meter (DCUU, DC5508U), with the cold side of film suspending in the air. The spot welding was used to connect Pt-Rh wires with the films, and the output properties of the thermoelectric generator composed of 5 pieces of films were collected by a multimeter (Agilent 34970A) and a constant current source (Agilent B2901A)²⁹. The bending test was conducted via a hand-cranked tensile testing machine (AIGU, ZP-1000).

First-principles calculations

This work utilized periodic slab models for all DFT calculations, conducted via the Vienna ab initio simulation package, including DFT-D3 empirical van der Waals corrections. The HSE exchange-correlation functional was applied within the generalized gradient approximation GGA³⁰. Electron-ion interactions were described using the projector-augmented wave method, employing a plane-wave basis set with a cutoff energy of 500 eV. A 3 × 4 × 3 Monkhorst-Pack k-point mesh for perfect bulk β-Ag₂Se with lattice constants a = 9.17 Å, b = 7.04 Å, c = 7.58 Å. The coverage criteria for energies and forces were set to 10⁻⁵eV and 0.02 eV/Å. The first order Methfessel–Paxton method with a smearing width of 0.05 eV was used. The model for the Ag-rich nanoprecipitate was constructed by removing one Se atom from the structure of Ag₁₆Se₈. For the (11-2) crystal plane of β-Ag₂Se and its derived models, a 3 × 2 × 1 Monkhorst-Pack k-point mesh was applied. To calculate the deformation potential, the positions of the CBM for the structures with strain magnitudes ranging from −1% to 1% were calculated in increments of 0.5%. The energy difference between the CBM and the core energy level yields E_CBM. The slope of the variation of E_CBM with strain represents the deformation potential of the material.

Numerical simulation

The software for numerical simulation is Comsol Multiphysics 5.6. The surrounding air around the film was set to be natural convection, and the thermal resistance between the composite film and silver electrodes was ignored.

Single parabolic band (SPB) model for carrier transport characteristics

For the SPB model, the transport coefficients are expressed below:

Density-of-states effective mass (m_d):

$${m}_{{\rm{d}}}=\frac{{\left(\frac{3{h}^{3}}{8\pi }{n}_{{\rm{H}}}\frac{\left(2r+\frac{3}{2}\right)}{{\left(r+\frac{3}{2}\right)}^{2}}\frac{F\, \left[2r+\frac{1}{2},\eta \right]}{{F\, \left[r+\frac{1}{2},\eta \right]}^{2}}\right)}^{\frac{2}{3}}}{2{m}_{{\rm{e}}}{k}_{{\rm{B}}}T}$$

(1)

Deformation potential (E_def):

$${E}_{{\rm{def}}}=\frac{{\frac{e{h}^{4}}{16\sqrt{2}{\pi }^{3}{\left({k}_{{\rm{B}}}T\right)}^{\frac{3}{2}}}\frac{{v}_{{\rm{l}}}^{2}d}{{{{\mu }_{{\rm{H}}}}({m}_{{\rm{d}}}{m}_{{\rm{e}}})}^{\frac{5}{2}}}\,\frac{\left(2r+\frac{3}{2}\right)}{{\left(r+\frac{3}{2}\right)}^{2}}\frac{F\, \left[2r+\frac{1}{2},\eta \right]}{{F\, \left[r+\frac{1}{2},\eta \right]}^{2}}\left. \right)}^{\frac{1}{2}}}{e}$$

(2)

Seebeck coefficient (S):

$$S=\frac{{k}_{B}}{e}\left(\frac{\left(\frac{5}{2}+r\right)}{\left(\frac{3}{2}+r\right)}\frac{F\left[r+\frac{3}{2},\eta \right]}{F\left[r+\frac{1}{2},\eta \right]}-\eta \right)$$

(3)

Hall Carrier concentration (n_H):

$${n}_{{\rm{H}}}=\frac{8\pi }{3}{\left(\frac{2{m}_{{\rm{d}}}{k}_{{\rm{B}}}T}{{h}^{2}}\right)}^{\frac{3}{2}}\frac{{\left(r+\frac{3}{2}\right)}^{2}}{\left(2r+\frac{5}{2}\right)}\frac{{F\left[r+\frac{1}{2},\eta \right]}^{2}}{\left[2r+\frac{1}{2},\eta \right]}$$

(4)

Hall Carrier mobility (μ_H):

$${\mu }_{{\rm{H}}}=\frac{e{h}^{4}}{16\sqrt{2}{\pi }^{3}{\left({k}_{B}T\right)}^{\frac{3}{2}}}\frac{{v}_{{\rm{l}}}^{2}d}{{({E}_{{\rm{def}}}*e)}^{2}{({m}_{{\rm{d}}}{m}_{{\rm{e}}})}^{\frac{5}{2}}}\frac{\left(2r+\frac{3}{2}\right)}{{\left(r+\frac{3}{2}\right)}^{2}}\frac{F\left[2r+\frac{1}{2},\eta \right]}{{F\left[r+\frac{1}{2},\eta \right]}^{2}}$$

(5)

Electrical conductivity (σ):

$$\sigma=e{n}_{{\rm{H}}}{\mu }_{{\rm{H}}}$$

(6)

Power factor (PF):

$${PF}={S}^{2}\sigma$$

(7)

In the above equations, the integral F [n, η] is defined by: $F[n,\eta ]={\int }_{0}^{\infty }[\frac{{x}^{n}}{1+{\rm{Exp}}(x-\eta )}]$. h, m_e k_B, e, and r are the Planck constant, the mass of free electron, the Boltzmann constant, the charge the electron, the scattering factor (r = −0.5 for acoustic phonon scattering). v_l is the longitudinal speed of sound of β-Ag₂Se (measured to be around 2540 m/s in this work), d is·the density of sample (the theoretical density of 8.24 g cm⁻³ was applied for calculation).