Subamorphous Thermal Conductivity of Crystalline Half-Heusler Superlattices

ABSTRACT The quest to improve the thermoelectric figure of merit has mainly followed the roadmap of lowering the thermal conductivity while keeping unaltered the power factor of the material. Ideally an electron-crystal phonon-glass system is desired. In this work, we report an extraordinary reduction of the cross-plane thermal conductivity in crystalline (TiNiSn):(HfNiSn) half-Heusler superlattices (SLs). We create SLs with thermal conductivities below the effective amorphous limit, which is kept in a large temperature range (120–300 K). We measured thermal conductivity at room temperature values as low as 0.75 W m−1 K−1, the lowest thermal conductivity value reported so far for half-Heusler compounds. By changing the deposition conditions, we also demonstrate that the thermal conductivity is highly impacted by the way the single segments of the SL grow. These findings show a huge potential for thermoelectric generators where an extraordinary reduction of the thermal conductivity is required but without losing the crystal quality of the system


Introduction
Understanding of heat propagation and the ability to tune the thermal properties constitute a topic of continuous and active research motivated by the increasing importance of thermal management and ways to recover waste heat energy as it is the case for the thermoelectric industry. This renewed interest in thermal management has introduced a number of novel concepts and ideas including: thermocrystals [1], thermal-cloaking, -transistors, -diodes and -memories [2][3][4][5][6][7][8][9], phonon-mean-freepath spectroscopy [10,11], among others.
The control of phonon propagation, the main heat carriers in semiconductors and insulators, is a crucial requirement for thermoelectric generation. Ideally, a material with thermal properties of an amorphous state (phonon glass) and electronic properties associated with good single-crystal semiconductor (electron crystal) are desired [12]. Materials' very low thermal conductivity, k, is also needed in other applications such as: thermal barrier coatings for gas turbine engines and thermal data storage devices [13].
The lowest k in crystalline systems is achievable through alloy scattering or the so-called alloy limit. But, the introduction of an extra scattering mechanisms, e.g., nanostructures, can exceed this limit. The use of superlattices (SLs) [14][15][16][17] and embedded nanoparticles [18] have demonstrated to be a good way to reduce k below the alloy limit, while maintaining the crystal quality of the material. The introduction of more and more scattering events can reduce even further this limit reaching its second minima: the amorphous limit. In this context, recent experiments showed that, by introducing small-periods in SLs, ultralow k values below the amorphous limit can be achieved [19][20][21]. Costescu et al. [19] and Pernot et al. [20] measured cross-plane thermal conductivity values (k ⊥ ) below the amorphous limit of Al 2 O 3 and Si in Al 2 O 3 :W and SiGe:Si SLs, respectively. Niemelä et al. also overtook the amorphous limit of TiO 2 using organic-inorganic (TiO 2 ):(Ti-O-C 6 H 4 -O) SLs. Moreover, Chiritescu et al. [22] also measured ultralow k ⊥ in layered WSe 2 thin films. k ⊥ values below the amorphous limit of WSe 2 crystal were achieved by controlling both order and disorder in the thin films.
In SLs, thus, it is natural to think that the smaller the period length (L), the smaller k ⊥ . However, several theoretical [15,[23][24][25] and experimental [17,26,27] reports have shown that for very thin L, the k ⊥ increases. In such limit, phonons experience the material as if it was composed of enlarged unit cells given by the size of L. The SL is seen as one homogenous material and the phonon transport is considered coherent [28]. The transition between coherent-incoherent (wave-particle) transport is observed as a minimum in the thermal conductivity, k ⊥ as a function of L [17,26]. This effect comes from the competition between phonons diffusively scattered at each interface and the band-folded ones. The first unambiguous experimental demonstration of this crossover was presented by Ravichandran et al. [17] using epitaxial perovskite-based SLs. Another fingerprint of coherent thermal transport was proposed by Luckyanova et al. [28] namely, a linear dependence of k ⊥ with respect to the number of periods as indicator of coherent thermal transport through the SL. This arises when the phonon mean free paths (MFPs) are equal to the total thickness of the SL (d) leading to the linear dependence of k ⊥ on the number of periods.
In any case, either by looking at the minimum or the linear dependence of the k ⊥ with respect to L, to observe coherent thermal transport it is necessary that the incoming thermal wave retains its phase after it has been reflected or transmitted across the interface. This implies that the scattering mechanisms at each interface should not be purely diffusive, otherwise, the interfacial roughness or intermixing will destroy phonon coherence and phase information will be lost [29,30]. Therefore, the presence of atomically smooth interfaces becomes mandatory. However, this last point is not fully understood. Recently, numerical simulations carried out by Qiu et al. found the same about linear dependence of k in rough periodic and aperiodic Si:Ge SLs with period thickness L = 20 nm [31]. These findings were associated to the low interface densities (1/L = 0.05 nm −1 ) and weak disorder scattering. In this case, the dominant thermal phonons are not affected by the disorder scatterings and they can transverse ballistically the SLs regardless of aperiodicity or interface roughness. Similar results were found by Wang et al. [32] and Chakraborty et al. [33] in rough periodic SLs and random multilayer (RML) structures made of artificial atoms. Both simulations showed the same linear-like behavior of k ⊥ vs L. However, the absence of a minimum in k ⊥ as a function of total thickness in the simulations performed by Wang et al. suggest a ballistic phonon transport rather than coherent effects [32].
In this work we report ultralow thermal conductivity in rough (TiNiSn):(HfNiSn) (with abbreviation (TNS):(HNS)) half-Heusler (HH) SLs. The period length of the SLs has been chosen to match crossover from incoherent to coherent transport in HH SLs [34]. The measured k ⊥ showed values below the amorphous limit of the effective material. As far as we know, these results are the lowest experimental values reported so far for any kind of HH compounds.

Previous results
The HH compounds investigated here are n-type narrow-band-gap semiconductors with quite large Seebeck coefficient and electrical conductivity [35,36]. However, the relatively high thermal conductivity still limits their thermoelectric performance and, hence, the industrial commercialization. For this reason, our previous studies were focused on the k reduction through SL structuration. We designed three different experiments to study the impact of the period length [34,37] and the period composition [38] on the electrical and thermal properties of HH SLs. Our findings revealed a room temperature crossover from incoherent to coherent thermal transport in HH SLs. The k ⊥ vs L exhibited a continuous diminution of k ⊥ as L decreases, showing a minimum of k ⊥ = 1.11 ± 0.06 W K −1 m −1 at L~3.2 nm. At smaller L the k ⊥ rises up entering in the coherent regime [34,37].

Experimental results and discussion
In this work, we have taken a different experimental approach to study the heat transport trough the SL. Instead of fabricating smooth and defect-free SLs, we have deteriorated the quality of the interfaces by changing the deposition conditions. We used DC magnetron sputtering to fabricate eight (TNS):(HNS) SLs with period thicknesses ranging 2.9 nm < L < 4.8 nm. The L was determined from the best fit of the X-ray diffractograms using CADEM: calculate X-ray diffraction of epitaxial multilayers [39]. Five SLs were grown with the same number of periods N = 37 (S 1 , S 2 , S 3 , S 7 and S 8 , respectively). Other three samples were deposited with different number of periods N = 111 (S 4 , S 6 ) and 148 (S 5 ). All these samples, except S 1 (homogeneous-growth), were grown 30 mm away from the center of the cathodes in the inhomogeneous part of the plasma cloud (inhomogeneous-growth). Two different deposition conditions were used here. S 1 , S 2 , S 6 , S 7 and S 8 were grown at low gas pressure and cathode power (low rate), while S 3 , S 4 and S 5 were grown at high gas pressure and cathode power (high rate). The surface roughness was determined from the root mean square of a two dimensional power spectral density plot of the sample surface measured by atomic force microscopy, AFM. For convenience, the AFM surface-roughness will be referred simply as roughness (η). Table 1 lists a summary of all the samples measured in this work. A detailed description of the sample fabrication can be found in the supporting information.
A cross-sectional transmission electron microscope (TEM) image of one SL with a roughness of η = 5.9 nm and period thickness L = 4.5 nm is displayed in Figure 1a. As it is displayed in the inset of Figure 1a, there is an intermixing of the SL layers, however, the SL still keeps the crystal and epitaxial quality as shown in the rocking curve in Figure 1b and its inset. The rocking curves reveal the broadening of a given diffraction peak. Defects such as mosaicity, atomic intermixing dislocations, among others, lead to spreading of crystal planes and thus a broadening of the linewidth [40]. In addition, the presence of the (002) and (004) film reflections around 2θ = 30º and 60º, respectively, confirm the crystallinity of all the samples discarding amorphization of the crystal structure (see Table 1. Summary of the deposition parameters (Ar pressure and cathode power), number of periods, period length (L), surface roughness (η), full width half maximum of rocking curve (FWHM,) total SL thickness and sample holder of investigated samples.  Figure S3, S4 and S6 in the supporting information). The crystal quality can also be appreciated in the high resolution TEM image, where it is possible to observe the well-ordered crystal structure (see inset Figure 1a). The cross-plane thermal conductivity was measured using well-known three-omega (3ω) method [41,42] in the differential configuration [43,44].
First, we focused on S 1 and S 2 , grown under the same low sputtering rates but at different distance of the cathodes. The homogeneously grown SL (S 1 ) shows significantly higher k ⊥ than the inhomogeneous SL (S 2 ) above 120 K. It appears that the difference in period lengths (L 1 = 2.9 nm and L 2 = 3.5 nm for S 1 and S 2 , respectively) may explain this finding. However, in our previous work, we found that the k ⊥ decreases as period length decreases achieving a minimum value k ⊥ ≈ 1.11 W K −1 m −1 at L ≈ 3.2 nm [34]. Then, as both period lengths of the SLs are located around this minimum, the k ⊥ should be almost identical. Therefore, the difference in k ⊥ cannot be associated exclusively to SL period. At first glance, the increase of η may also explain this behavior. Moreover, after change in the deposition conditions to induce higher η (S 3 ), the k ⊥ decreases even more reaching values as low as the theoretical amorphous limit of HNS and below the amorphous limit    of an effective material [45]. This behavior is preserved along a wide temperature range 100 < T < 300 K as it is displayed in Figure 2a. A deeper description of the theoretical amorphous limit is given in section 3 of the supporting information. As we already discarded a possible amorphization of the films and the impact of L, the other factor that could explain the very low k ⊥ is a higher concentration of defects and the loss of the epitaxial growth of S 3 . But, the sharpness of the rocking curve of S 3 (Γ 3 = 1.08º) in comparison with S 1 (Γ 1 = 1.31º) and S 2 (Γ 2 = 1.51º) indicates that S 3 is superior in terms of epitaxial quality to the other two SLs and the k ⊥ of S 3 should be even higher. Thus, the extremely low k ⊥ values shown by S 3 can be directly related to the high increase of η. But, for the case of the other two SLs, as the FWHM of Γ 1 < Γ 2 , it is reasonable that the difference in k ⊥ can be associated to a combination of both the loss of crystal quality and the increase of roughness. This dependence is better appreciated in Figure 2b, where k ⊥ is plotted as a function of η and Γ/η (bottom and top x-axis, respectively). We can see that k ⊥ monotonically decreases with η from 300 K to 170 K. While for lower temperatures, we observe that k ⊥ is getting constant for low η values. On the other hand, we can observe that for high roughness (or small FWHM/η) the k ⊥ increases dramatically. Similar behavior was also observed by Termentzidis et al. [46] and Merabia and Termentzidis [47] using molecular dynamics simulations. We will return to this point later. Now, we analyze behavior of k ⊥ as the number of periods increases for the SLs grown with similar conditions as S 3 . Figure 3a shows the k ⊥ as function of number of periods corresponding to the samples S 3 , S 4 and S 5 , respectively. The period length in this case should be equal but it is possible to see a shift of the satellite peaks ( Figure S4 supporting information). The calculated period length was found L = 3.9 nm and 4.8 nm for S 4 and S 5 , respectively.
From Figure 3a, one sees that k ⊥ rises continuously at 250 K and at 170 K similarly to the behavior observed by Luckyanova. 30 While for 100 K the k ⊥ seems to be constant for thicker samples. The nearly linear dependence k ⊥ on the number of periods seems to indicates that part of the heat is transported still by phonons with mean free path in the order of the sample thicknesses. Other interpretation of this phenomena can be also associated to epitaxial quality of the thicker samples.
The inset of Figure 1b shows very sharp FWHM S 4 (Γ 4 = 0.77º) and S 5 (Γ 5 = 0.75º). This means that both S 4 and S 5 have superior epitaxial quality than S 3 . As this effect may also play a role here, the k ⊥ as function of FWHM is plotted in Figure 4a. From this graph it is possible to observe certain correlation between k ⊥ and the FWHM. Except for S 3 , there is a about-linear decrease of the k ⊥ as FWHM increases. Now, if we pay attention to the plot of the thermal conductance, k ⊥ · d (where d is the total thickness of each SL), vs FWHM there is a clear correlation between the samples grown under the same deposition conditions (see Figure 4b). The other important parameter that we have to take into account is the surface roughness, which rises significantly with the number of periods. In contrast to previous cases, here the surface roughness of S 4 and S 5 is η~28 nm, which is six times larger than the L of the SL. Such huge η should also impact on the k ⊥ reducing it even more, in spite of that, we observe that the experimental k ⊥ is still increasing (see Figure 2b). We can notice that the k ⊥ rises up for larger number of periods (S 6 , N = 111 and L = 4.5 nm).
As we mentioned above, theoretical simulations of Termentzidis et al. [46] and Merabia and Termentzidis [47] showed an increase of k ⊥ of SLs with very rough interfaces compared with atomically smooth SLs. They suggested that the thermal conductivity of a SL is mainly controlled by the Kapitza resistance of interfaces, which in turn seems to be governed by the interfacial area. It is because a large majority of phonons have wavelength (λ) larger than η and they see the interface as a planar one. Then, the transmitted heat flux is controlled by the projected area. On the other hand, for very rough interfaces, most of the phonons have much smaller λ than the η, then, the phonons will not feel the interface as planar, the phonon scattering will be incoherent and the transmitted heat flux is controlled by real contact area of the rough interface. In other words if λ > η, the effective scattering area would be the projected one and k ⊥ decreases slightly. For η~λ the interface will strongly scatter phonons and k ⊥ will decrease even further. But if η > λ the interfacial scattering becomes again negligible and the transmitted energy is proportional to the true area [47]. Similar behavior is also observed in SLs grown at low deposition rate (see Figure 3b).
Another interesting question concerning reduction of the thermal conductivity is, whether it is caused just by the fact that the one single component of the SL grows in a special way or if it is caused by the interaction of both components. To test this effect a double substrate holder was used for deposition. On this holder, there is room for two substrates lying side by side. So, during the deposition processes, one substrate is a little bit closer to the HNS and the other to the TNS, respectively. As the samples were grown in the inhomogeneous region of the plasma cloud, one of the SLs will contain a little bit more TNS or HNS per period than the other. This effect can be appreciated in XRD diffractograms displayed in the Figure S6 in the supporting information. The XRD diffractogram shows that the maxima of (002) HH peaks are shifted. As expected, the sample closer to HNS cathode (S 6 ) has a maximum at 29.4º which is closer to (002) peak of HNS (29.36º). While the sample closer to TNS cathode (S 7 ) has a maximum at 29.9º which is closer to (002) peak of TNS (30.06º). The rocking curves of both samples show an equal FWHM value of 1.21º. Here we focused on SLs having the same number of periods, which were grown with low sputtering rate to minimize the impact of the roughness. In Figure 3c, the k ⊥ of S 6 and S 7 are compared to the SL grown using single sample-holder S 2 . It is clear that the sample that contained more TNS (S 6 ) has higher thermal conductivity than the sample that contained more HNS (S 7 ) and the single sample-holder (S 2 ). Similar results were also observed in our previous investigation in HHs SLs. Where we found a minimum of the k ⊥~1 .39 W K −1 m −1 for SL having the same amount of each material [38].
Coming back to the idea of coherent transport, while it is interesting to speak about a possible coherent transport in rough SLs, we cannot proof that the heat transport is influenced by coherent phonons just based on the nearly linear dependence of k ⊥ on the number of periods. Other mechanism such as material intermixing, different degree of interface roughness and privileged growth of one of the constituent materials of the SL could also explain this linear dependence of k ⊥ . Therefore, an observation of a linear increase of k ⊥ with the number of periods does not necessarily imply coherent transport. Rather, it can be explained by phonon mean free paths larger than the system length, i.e., ballistic transport. Furthermore, it is also important to mention the impact of low and high growth rates on the structures and their thermal conductivities. While the low rate warranties very smooth interfaces with low surface roughness, it induces a mosaic effect reducing the thermal conductivity of the samples. On the other hand the high rate reduces the mosaic effect but leads to very rough interfaces. At the very rough interfaces, the interfacial scattering becomes negligible and the transmitted energy is favored [47].

Conclusions
In this work we found out that the samples that were grown in the inhomogeneous region of the deposition cloud exhibited significantly lower thermal conductivity than the sample grown at the homogenous part. The thermal conductivity can be reduced even below the amorphous limit by using higher gas pressure and cathode power for deposition process. This is an outstanding result because it means that a solid body with good crystalline qualities (as implied by the quite narrow rocking curves) has a lower thermal conductivity than it should have in amorphous state. We also observed experimentally a linear-like increase of k ⊥ as a function of the number of periods for SL grown under variable deposition conditions. While this behavior has been reported before as coherent transport, we cannot prove that this is the case in this work. Other parameters such as the degree of intermixing, interface roughness and crystal quality may also play a role. Furthermore, we have also demonstrated that the thermal conductivity is influenced by the way in which one of the single components grows within the inhomogeneous region. Finally, our findings show a large potential for thermoelectric generators where a huge reduction of k is required but without losing the crystal quality of the system.