Suppressing thermal conductivity of nano-grained thermoelectric material using acoustically hard nanoparticles

Intensive research has been undertaken to improve thermoelectric (TE) properties. One of the most important applications is thermoelectric cooling, with particular interest at a temperature of around 300 K and below.¹ Thermoelectric cooling has many advantages, such as compact size, portability, fast response, and with no moving parts. These make it an attractive option for thermal management for sensors and electronics, such as lasers² and computer chips,³ cryogenic cooling of infrared detectors and superconductive devices,⁴ as well as for personalized thermoregulation.⁵ However, more wide-spread application of TE cooling and refrigeration is limited by the low-energy conversion efficiency or the coefficient of performance (COP), which is dictated by the thermoelectric figure of merit, defined as $ZT=S2σT/κ$⁠, where S is the Seebeck coefficient, $σ$ is the electrical conductivity, T is the temperature, and $κ$ is the thermal conductivity.⁶ 

One of the extensively exploited approaches to improve the ZT in the literature is to introduce nano-grained structures.^7–10 The high-density grain boundaries enhance the scattering rate of phonons, the primary heat carriers in most semiconductor TE materials, and thus can effectively reduce the lattice thermal conductivity $(κlat)$ without considerable degradation in the power factor $(S2σ)$⁠. This method has shown to be effective in significantly improving the ZT at the room and high temperature range, for instance, in p-type Bi_0.5Sb_1.5Te₃,⁹ n-type Bi₂Te_2.7Se_0.4,¹¹ and n and p- type SiGe alloys.^12,13 However, the ZT improvement is much smaller at a low temperature range (≤300 K).^9,14–16 This is because the grain boundary is not effective in scattering phonons with a long mean free path (MFP), which tend to dominate the thermal transport at low temperature. For instance, Wang et al.¹⁷ found that the phonon transmission through the grain boundary depends on their frequency or MFP. The low frequency (or long MFP) phonons have a higher probability of transmitting through the grain boundaries than the high frequency phonons.^17,18 This was also shown in ballistic phonon penetration through a thin amorphous silica layer.¹⁹ Therefore, the nano-grained structure may not effectively scatter the low frequency phonons. It is still challenging to enhance the scattering rate of these long MFP phonons at around 300 K or below. Recent work on nanograins with dense dislocation arrays in BST showed the great promise of engineering the grain boundary to effectively scatter low frequency phonons and lower the thermal conductivity.²⁰ However, there is still a lack of a generic approach to additionally suppress the MFP of low frequency phonons beyond what is offered by the nanograins in thermoelectric materials.

We seek to develop a generic strategy to suppress $κlat$ in nano-grained TE materials at low temperature. Our approach is based on embedding nanoparticles (NPs) into nano-grained materials to introduce a stronger scattering effect for phonons, especially those with longer MFPs. There is plenty of prior work on embedding NPs into nano-grained materials.^21–26 However, most of these reports were predominantly focused on ZT values, especially at high temperature (>300 K) for energy harvesting applications. For example, Dou et al.²¹ mixed 40 nm SiO₂ NPs with Bi_0.4Sb_1.6Te₃ powders and found ∼20% improvement of ZT at 363 K, which was attributed to both enhanced power factor and suppressed thermal conductivity. However, the work of Dou et al. did not study the SiO₂ NP effect at temperature below 300 K, where low frequency phonons are expected to be more dominant in $κlat$⁠. There is also a lack of a systematic study on the effect of NP type on thermal conductivity. From the theoretical point of view, it is expected that NPs having a greater contrast in the acoustic impedance with the matrix materials would exhibit a larger interfacial resistance and hence lower thermal conductivity. This concept, while extensively exploited in systems such as multilayers,²⁷ has not been studied in-depth in nano-grained TE materials with NP inclusions.

In this paper, we investigated the NP effect on $κlat$ in the nano-grained materials at low temperature $(≤300K)$⁠. We mixed 10 nm SiO₂ NPs and 15 nm diamond NPs with nano-powders of Bi_0.5Sb_1.5Te₃ (referred to as “BST” hereafter), a prototypical p-type thermoelectric material. We observed strong reduction of thermal conductivity in the samples mixed with both types of NPs. Furthermore, we found that the diamond/BST samples showed larger reduction in $κlat$ compared to that of the SiO₂/BST samples with the same NP volumetric concentrations. This difference can be attributed to the larger interfacial acoustic mismatch of the diamond/BST than that of the SiO₂/BST. The thermal boundary resistance of diamond/BST was found to be about 10 times higher than that of the SiO₂/BST. Because of the strong suppression of $κlat$⁠, we observed 15% improvement of ZT in 0.5 vol. % diamond/BST at room temperature compared to the pristine nano-grained BST.

Figure 1(a) shows the schematic of the NPs embedded in the BST matrix. The embedded NPs are distributed around the nano-grained boundaries. The long MFP phonons reaching the grain boundary will be scattered by the NPs. This can effectively reduce the MFP of these phonons, thus causing additional reduction in the $κlat$of nano-grained materials. The starting TE materials are small ingots of p-type BST, which were also used in our earlier work.^28,29 The BST nano-powders were obtained by high energy ball milling (SPEX 8000 M mixer/mill) for 10h in an Ar atmosphere. The SEM images [Figs. 1(b) and 1(c)] of the nano-powders after ball milling show that the majority of the particles are less than 100 nm in diameter. The BST nano-powders were mixed with the target nanoparticles (SiO₂ or diamond) at 0.5%, 1%, and 5% volumetric concentrations and ball milled for additional 4h in an ethanol solution in the Ar atmosphere. The wet mixing method was used to avoid possible agglomeration of NPs before and during mixing with BST nano-powders. Figures 1(d) and 1(e) show TEM images of SiO₂ NPs and diamond NPs, with diameters of 10 and 15 nm, respectively. After the mixing, the milling jar was moved into a glovebox filled with Ar, with an O₂ level controlled to be less than 0.5 ppm. The mixed powders were dried and pre-pressed at room temperature into the graphite die in the glovebox. Then, the cold-pressed pellet was further hot pressed in another glovebox with the pressing condition of 400 °C and 60 MPa, for 3 min, to yield a dense disk-like sample. Note that transition of diamond to graphite requires very high temperature heat treatment (600 °C in an oxygen environment and up to 1900 °C in an oxygen-free environment) due to the large energy barrier between diamond and graphite.^30–32 Therefore, the diamond NPs in our samples are expected to remain in the diamond phase under our hot-pressing conditions (400 °C) in a nearly oxygen-free environment.

The samples after hot pressing are of a disk-like shape with 12.7 mm diameter and 2 mm thickness, as shown in Fig. 2(a). Figure 2(b) shows the x-ray diffraction (XRD) data for the BST samples mixed with NPs at different volumetric ratios. All the peaks correspond to the BST phase; peaks form SiO₂ and diamond are not identifiable likely because of the small size and low volume concentration of these phases. The average grain sizes, which are summarized in Table I, were estimated from the XRD data using the Scherrer equation,³³ 

where $λ$ is the wavelength $(λ=0.154nm)$⁠, K is the shape factor (taken as 0.90, a good approximation commonly used for BST³⁴), $β$ is line broadening at half of the maximum intensity (FWHM) that is corrected with the consideration of the broadening from the XRD machine, and $θ$ is the Bragg angle.

Three samples (pristine, 5.0 vol. % SiO₂ NPs, 5.0 vol. % diamond NPs) were selected for further microstructure analysis using scanning electron microcopy (SEM) and transmission electron microscopy (TEM). The details of these characterization methods are shown in the supplementary material.

The samples were then polished into a regular cuboid shape (2 × 2 × 10 mm³), with the long axis being the direction perpendicular to the hot-pressing direction. The sample densities were measured using Archimedes’s method and were listed in Table I. Theoretical density for the samples containing NPs was calculated using a rule of mixtures of the appropriate volume percent and the densities of BST, diamond, and SiO₂. The thermoelectric properties of the samples, including thermal conductivity, electrical conductivity, and Seebeck coefficient, were measured with a physical property measurement system (PPMS, Quantum Design, Inc., San Diego) within the temperature range of 50–300 K for the pristine and SiO₂ NP mixing samples and 70–300 K for the diamond NP mixing samples (these samples were measured later when the PPMS could not stably cool below 70 K). All the transport properties were measured along the long axis direction. The measurement errors were estimated automatically in the software of PPMS, considering data curve fitting residue, heater power error, heat loss, and temperature error.

Table I shows the average grain size obtained from the XRD analysis. The grain sizes are generally within the range of 20–40 nm, which are typical for ball-milled BST samples.⁹ We also note that the XRD patterns are similar among all the samples, even with 5 vol. % of the SiO₂ or diamond NPs. Even though one could generally expect the possible observation of additional peaks from the NPs at such a concentration, we note that there are several factors that could have prevented us from observing these peaks: amorphous nature of the SiO₂ NPs, the small crystallite size of the diamond NPs, and the short penetration depth of the x ray due to the high Z number of the BST matrix.

From the SEM and TEM results shown in the supplementary material, we can confirm the inclusion and uniform distribution of the SiO₂ and diamond NPs in the BST matrix and the small crystallite size of the BST matrix, which is consistent with the results obtained from the XRD. However, it is very challenging to directly observe the individual NPs embedded in the BST matrix from the SEM or TEM. The results also suggest some agglomerated particles and various degrees of dispersion of the particles. This suggests that the particle distribution is likely deviated from the ideal situation schematically shown in Fig. 1. Nevertheless, given the same sample preparation methods among the three types of the samples, namely, pristine, SiO₂ NPs, and diamond NPs and the similar distribution of the SiO₂ and diamond NPs, we can suggest that differences in the thermal and thermoelectric transport properties can be attributed to the NP inclusions.

Figure 3(a) shows the measured temperature dependent thermal conductivity for BST mixed with SiO₂ and diamond NPs with different volumetric ratios. Comparing with the BST samples without NP mixing (or pristine BST), i.e., 0 vol. % NPs, the samples mixed with SiO₂ or diamond NPs show large reduction in thermal conductivity. At 300 K, the thermal conductivity values are 1.35, 0.77, and 0.67 W/m K for pristine BST, 5 vol. % SiO₂/BST, and 5 vol. % diamond/BST, respectively. The reduction is even more significant at low temperature. At 70 K, the thermal conductivity decreased from 1.92 W/m K for the pristine nano-grained BST down to 0.91 W/m K for 5 vol. % SiO₂/BST and 0.75 W/m K for 5 vol. % diamond/BST, representing 52.6% and 60.9% reduction, respectively.

The measured thermal conductivity $κtotal=κe+κbp+κlat$⁠, where $κe$⁠, $κbp$⁠, and $κlat$ are majority carrier electronic, bipolar electronic, and lattice thermal conductivity, respectively. For (BiSb)₂Te₃ materials, Kim et al.³⁵ showed that $κbp$ is negligible at T < 300 K. Therefore, the total thermal conductivity can be approximated as $κtotal=κe+κlat$⁠. The $κlat$ can thus be obtained by subtracting $κe$ from $κtotal$⁠, as shown in Fig. 3(b). Here, the $κe$ is calculated based on the Wiedemann–Franz law, i.e., $κe=LσT$⁠, where $σ$ is the electrical conductivity, T is the temperature, and L is the Lorenz number, which is calculated self-consistently using the electrical conductivity and Seebeck coefficient data, as we shall discuss in detail in Sec. E. Figure 3(b) shows that the $κlat$ was significantly reduced by embedding NPs into the BST matrix. Relative to the pristine nano-grained BST, the reduction of the $κlat$ for SiO₂/BST at 0.5 and 5 vol. % are 12% and 39% at 300 K, respectively. The reduction percentage is larger for the diamond/BST, 23% and 54% for 0.5 and 5 vol. % at 300 K, respectively.

In order to better understand the scattering mechanisms and the effect of NPs on the phonon MFP spectrum, we used a phenomenological model for the $κlat$ as^17,20,36,37

where c is the volumetric specific heat; index number i = 1, 2, 3 represent two transverse and one longitudinal phonon modes, respectively; v is the speed of sound; $kB$ is the Boltzmann constant; $ℏ$ is the reduced Planck's constant; and $x=ℏωkBT$⁠. $Λeff$ is the effective mean free path of phonon,^17,37

where $Λd−1=Aω4/νs$ is for defect scattering,^38,$ΛU−1=Bω2T/νi$ for Umklapp scattering,^39,$Λb−1=[C(ω0/ω)Dg]−1$ for frequency-dependent boundary scattering,¹⁷ and $Λnp−1=Nnpσnp$ for nanoparticle scattering.^40,41$Dg$ is the grain size obtained from XRD (Table I), $ω0$ is the cut-off frequency,^42,43 “A,” “B,” and “C” are adjustable parameters, $Nnp=f/(16πDnp3)$ is the volume concentration of NPs, f is the volume concentration ratio, $σnp−1=σs−1+σl−1$ is the effective scattering cross section, and $σs$ and $σl$ are scattering cross sections for short and long wavelength phonons, given as^40,44

where $δ=(Δρ/ρ)2+3(2Δν/ν)2$⁠, $Δρ/ρ$⁠, and $Δν/ν$ are differences of density and speed of sound, respectively, $q=ω/ν$ is the phonon wave vector, and “E” is an adjustable parameter.

To fit the model with the experimentally extracted $κlat$ in Fig. 3(b), we first adjusted the “A,” “B,” and “C” with $f=0$ for the pristine sample and obtain the optimal parameters: $A=2.40×10−41s−1rads−4$⁠, $B=1.10×10−17s−1K−1rads−2$⁠, and C = 0.58. Next, we considered the NP scattering effect for different NP concentrations. The diameters of SiO₂ and diamond NPs are 10 and 15 nm, respectively, and the volumetric concentration f varies from 0.5% to 1% to 5%. By fixing the “A,” “B,” and “C,” we adjusted the parameter “E,” and found the optimal fitting results with E of 9 and 32 for the SiO₂ and diamond mixing samples, respectively. Note that we used the same “E” value for the same type of NPs at different volumetric concentrations. The model fits well for both types of NPs at different volumetric concentrations, as shown in Fig. 3(b).

Since we observed significant reduction of $κlat$ in the NP mixed samples in Fig. 3(b), it would be interesting to investigate the MFP distribution of phonons in these materials. After fitting the temperature dependent $κlat$⁠, we kept the same fitting parameters and calculated the corresponding MFP distribution of $κlat$⁠. The MFP distribution can be modeled by transforming the integration in Eq. (2) from phonon frequency $ω$ to phonon MFP, following the procedure developed by Yang and Dames.⁴⁵ Figure 4(a) shows the MFP distribution for BST samples with and without NPs at 300 K. The MFP distribution of the NP embedded samples is shifted toward a shorter MFP range compared to that of the pristine sample. The maximum phonon MFP reduces from 200 nm in the pristine sample down to 30 nm in the 5 vol. % diamond/BST. This clearly demonstrates that the embedded NPs can effectively scatter the long MFP phonons, thus reducing the $κlat.$ Furthermore, the MFP reduction of the diamond/BST is stronger than that of the SiO₂/BST with the same NP volumetric concentrations. For example, phonons with MFP longer than 10 nm contribute 24% of $κlat$ in 0.5 vol. % SiO₂/BST, but these long MFP phonons only contribute 13% of $κlat$ in 0.5 vol. % diamond/BST.

We also investigated the low temperature effect on the phonon MFP distribution, as shown in Fig. 4(b). We compared the MFP distribution for the pristine sample, 0.5 vol. % SiO₂/BST and 0.5 vol. % diamond/BST samples at 300 K and 50 K, and found that phonons with long MFP contribute more significantly to $κlat$ at 50 K than that at 300 K. For example, phonons with MFP larger than 10 nm contribute to 31% of $κlat$ at 300 K for the pristine sample. When the temperature decreased to 50 K, this ratio increased to 57%. Since the embedded NPs dominantly scatter the long MFP phonons, the reduction of $κlat$ by the NPs is more significant at low temperature than at high temperature. This is consistent with the lattice thermal conductivity results shown in Fig. 3(b).

The fact that diamond/BST samples showed lower thermal conductivity than the SiO₂/BST samples is counter-intuitive due to two reasons: (1) diamond has thermal conductivity (>2000 W/m K) of three orders of magnitude higher than that of SiO₂ (1.4 W/m K) and (2) SiO₂ NPs have a slightly smaller NP diameter in the samples than the diamond NPs. The different scattering strength of SiO₂ and diamond NPs can be understood from the interface thermal resistance between NPs and the BST matrix. Table II shows the mass density and speeds of sound of BST, SiO₂, and diamond.^37,46 Within the framework of a diffuse mismatch model (DMM), the thermal boundary resistance between media 1 and 2 (e.g., SiO₂ and BST or diamond and BST) can be calculated as^46,47

where “j” is the phonon mode number, v is the speed of sound, $ℏ$ is Planck's constant, and $f0$ is the Bose–Einstein distribution function. $α1→2$ is the transmission probability, which is defined as

where $δω,ω′$ is the Kronecker delta, v is the speed of sound for media 1 or 2, and DOS is the density of states of media 1 or 2.

The calculated thermal boundary resistance $(Rbd)$ of SiO₂/BST and diamond/BST at 300 K is listed in Table III. The $Rbd$ of diamond/BST is $15.39×108K⋅m2W$⁠, about 10 times larger than that of SiO₂/BST $(1.51×108K⋅m2W)$⁠. The larger $Rbd$ in diamond/BST is originated from the larger acoustic mismatch in diamond/BST compared to SiO₂/BST, as shown in Table III. This explains the stronger scattering of the embedded diamond NPs in BST. In our theoretical model above, we have a fitting parameter “$E$_,” which represents the effective scattering strength of the NPs to the phonons. The “$E$” value of the diamond/BST is ∼3.5 times as large as that of the SiO₂/BST, consistent with the fact that diamond/BST has higher $Rbd$⁠.

Electronic thermal conductivity $(κe)$ that we used to subtract from $κtotal$ was modeled with the Wiedemann–Franz law $(κe=LσT)$⁠, which requires electrical conductivity $(σ)$⁠, Lorenz number (L), and temperature (T). Figure 5(a) shows the measured $σ$ as a function of temperature for different samples. The Lorenz number can be calculated with the single parabolic band model,^48–50 which is obtained by solving the Boltzmann transport equation,

where r is the scattering parameter (⁠$r=−12$ for acoustic phonon scattering) and $η$ is the reduced Fermi energy, which can be solved with the measured Seebeck coefficient [Fig. 5(b)],

where $Fj(η)=∫0∞ϵj1+exp(ϵ−η)dϵ$ is the Fermi integral.

The calculated temperature dependent Lorenz numbers were shown in Fig. 5(c). The Lorenz number shows strong temperature dependence, ranging from $1.6×10−8V2/K2$ at 300 K to $2.3×10−8V2/K2$ at 50 K.

From Fig. 5(a), we observed that the embedded NPs have a detrimental effect on the $σ$⁠. This may be caused by the difference in the work function of diamond (4.15 eV) or silica (5 eV) and that of the BST (>5.2 eV; work function of n-type Bi₂Te₃ is 5.1–5.31 eV^51,52 and that BST is large because it is p-type); thus, the electron transport is partially impeded by the embedded NPs. Previous study has shown that mixing NPs could lower the electrical conductivity.⁵³ On the other hand, the Seebeck coefficient in the NP mixing samples remains about the same as the pristine sample within the measurement uncertainty, as shown in Fig. 5(b). This indicates that the carrier concentrations are similar between the pristine and NP mixing samples. To further illustrate the effect of the NPs on the thermal and electrical transport, we calculated the ratio of the room temperature $σ∗$ and $κtotal∗$ of the NP mixing samples, where $σ∗$ and $κtotal∗$ are the electrical and thermal conductivity values normalized to those of the pristine samples. As shown in Table IV, this ratio is within 10% of 1.0 for all but one samples (0.88 for the 5% diamond sample). There is a general trend that this ratio decreases with increasing NP concentration, meaning the $σ$ decreases faster than the $κtotal$⁠, resulting in slightly lower ZT values in samples with high NP concentrations.

The significant reduction of $κlat$ can potentially improve the thermoelectric performance of the nano-grained materials. Figure 6 shows the ZT numbers for different samples at 50–300 K. We have observed 15% and 10% improvement of ZT in the 0.5 vol. % and 1 vol. % diamond NP embedded samples at 300 K. However, ZT is decreasing with further increasing diamond mixing ratio, which is mainly because of the reduction of electrical conductivity for samples with higher volumetric concentration, as shown in Fig. 4(a). The $κlat$ reduction is not as large in SiO₂/BST samples as that in diamond/BST, and we could not see appreciable improvement in ZT for the SiO₂/BST samples. For samples with higher mixing concentrations, the reduction in the power factor was similar to or larger than the reduction of thermal conductivity; thus, the ZT remained the same or slightly decreased compared with the pristine samples. Future work should be focused on maintaining the power factor while reducing the thermal conductivity by embedding acoustically hard NPs. Possible strategies include using NPs with a suitable potential barrier height with the TE matrix to minimize the charge carrier scattering or utilizing electrically conducting NPs that could potentially enhance the power factor via doping and/or low-energy electron filtering effects.^54–56 

In summary, we have demonstrated the strong reduction of lattice thermal conductivity in nano-grained BST by embedding 10 nm SiO₂ and 15 nm diamond NPs. The diamond NPs at 5 vol. % effectively reduced $κlat$ up to 54% at 300 K and 65% at 70 K compared to the pristine nano-grained BST. The reduction is mainly due to the strong scattering of the long MFP phonons by the large NPs. The MFP distribution of NPs/BST samples is shifted significantly toward a shorter range by embedding NPs. The phonons with MFP larger than 10 nm contribute 31% to $κlat$ in the pristine BST at 300 K, but this ratio decreases to 24% and 13% in 0.5 vol. % SiO₂/BST and 0.5 vol. % diamond/BST, respectively. SiO₂ and diamond NPs showed different scattering strengths for the long MFP phonons, which is due to the different interfacial acoustic mismatch between the NPs and the BST matrix. The thermal boundary resistance of diamond/BST is found to be about 10 times as large as SiO₂/BST. Due to the large reduction of $κlat$⁠, we have observed 15% and 10% improvement of ZT in the 0.5 vol. % and 1 vol. % diamond NP embedded samples at 300 K, respectively.

See the supplementary material for a microstructure analysis of selected samples.

The work done at UCSD and GE&R was supported in part by a grant from the U.S. Naval Research Laboratory (Award No. N00173-14-1-G016, on material synthesis, XRD, transport measurements, and modeling). The SEM and TEM work conducted by TI was supported by Ryukoku University. We thank Dr. Elizabeth Rubin for the help with the XRD.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors have no conflicts to disclose.