Precise control of thermal conductivity at the nanoscale through individual phonon-scattering barriers

Precise control of thermal conductivity at the nanoscale through individual...

(Parte 1 de 2)

Precise control of thermal conductivity at the nanoscale through individual phonon-scattering barriers

G. Pernot, M. Stoffel, I. Savic, F. Pezzoli, P. Chen, et al.*

The ability to precisely control the thermal conductivity (κ) of a material is fundamental in the development of on-chip heat management or energy conversion applications. Nanostructuring permits a marked reduction of κ of single-crystalline materials, as recently demonstrated for silicon nanowires. However, silicon-based nanostructured materials with extremely low κ are not limited to nanowires. By engineering a set of individual phonon-scattering nanodot barriers we have accurately tailored the thermal conductivity of a single-crystalline SiGe material in spatially defined regions as short as ∼15nm. Single-barrier thermal resistances between 2 and 4×10−9 m2 KW−1 were attained, resulting in a room-temperature κ down to about 0.9Wm−1 K−1, in multilayered structures with as little as five barriers. Such low thermal conductivity is compatible with a totally diffuse mismatch model for the barriers, and it is well below the amorphous limit. The results are in agreement with atomistic Green’s function simulations.

Accurately tailoring the thermal conductivity of nanostructured materials with high spatial resolution is a fundamental challengeformicro- andnanoelectronicsheatmanagement, and for micro/nanoscale energy conversion on a chip1–4. Previous work on nanoscale thermal transport has demonstrated that in some cases nanostructuring can reduce the thermal conductivity of a material below that of its disordered alloy counterpart5–7, and can even beat the amorphous limit8, which for a long time was believed to represent a bound to the minimum attainable thermal conductivity of a material with a given composition9. In dislocation-free SiGe/Si multilayered materials however, it has not been clear how low the thermal conductivity can be pushed. A plausible lower bound when the SiGe layers are very thin would be given by a model in which ballistic Si layers are separated by interfaces, or phonon barriers (the thin SiGe regions), where phonons are scattered in a completely diffusive way. This is the diffuse mismatch model (DMM) in the particular case of no acoustic mismatch between the two sides of the interface10. In general, however, previous works showed only a weakly diffusive behaviour of the interfaces, with layer resistances lower than those predictedbytheDMM.Whenthelayersarecomposedofnanodots, this may be due to the low areal fraction covered by the interface dots7. However, in the opposite limit of fully continuous SiGe interlayers, the room-temperature interface thermal resistance is also about three times lower than the DMM (ref. 6). This raises the question: is it possible to achieve highly diffusive interfaces in SiGe/Si systems? Besides, most previous measurements were carried out on systems above 1µm thick and comprising over a hundred periods. Thus, it was unclear whether much thinner systems would still preserve the individually additive character of the single interface resistance, or whether ballistic effects across multiple periods might occur, rendering the concept of thermal conductivity inadequate for such thin regions11.

Here we answer the two questions above, and show that: highly diffusive interfaces can be achieved in dislocation-free SiGe/Si nanodot systems; and because of this, a well-defined thermal

*A full list of authors and their affiliations appears at the end of the paper.

conductivity can be accurately tailored for material regions as short as ∼15nm, comprising just a small number of periods. Two independent measurement techniques (see the Methods section and Supplementary Information), heterodyne picosecond thermoreflectance (HPTR) and the differential 3ω method, were employed to evaluate the thermal conductivities, yielding results consistent with an atomistic Green’s function (AGF) simulation of thephonontransportthroughtheSiGenanodotbarriers. The structures studied here (see Fig. 1a) consist of 5 and 1 layers of epitaxial Ge nanodots separated by Si spacers with thickness tSi (see the Methods section). The first island layer was obtained by the deposition of about 6monolayers (ML) of Ge, leading to the formation of small {105} faceted islands on top of a 3–4-ML-thick wetting layer (Fig. 1c). Dots have an average height of 1.2±0.2nm and a surface density of ∼8 × 1010 cm−2, with a fractional area coverage of about 70%. In the upper layers, the Ge coverage was reduced to prevent the occurrence of misfit dislocations12, as verified by extensive transmission electron microscopy (TEM) investigations (see, for example, Fig. 1b). In comparison with most of the previous works7,13,14, our multilayers were grown at a lower substrate temperature (500 ◦C), resulting in smaller dots (‘hut-’ instead of ‘dome-’ and ‘pyramid-’shaped clusters) with a higher Ge contentandhighersurfacedensities.Fromatomicforcemicroscopy (AFM) measurements carried out on the topmost layer of the stack (see Fig. 1d) we find that the areal densities in different samples vary between 1.2 and 5 × 1010 cm−2, which is significantly larger than those explored so far (up to ∼7×109 cm−2 in ref. 7). Only in ref. 15 the same nominal growth temperature was used, but the study was limited to relatively large interlayer spacing (20nm) and low-density islands (7×109 cm−2).

To characterize the cross-plane thermal conductivity, κ, of the samples we have developed the HPTR approach16. Numerous authors mentioned17–21 that the standard homodyne configuration of the picosecond thermoreflectance technique introduces a number of artefacts in the experimental signal and a number of corrections need to be applied before a correct interpretation of

ARTICLES NATUREMATERIALSDOI:10.1038/NMAT2752 p-Si (001) 100 nm Si buffer

Si spacer tSi = 3, 6, 9, 12 nm

12 nm

5 nm 4 10


Figure 1 | Sample structure. a, Schematic of the self-assembled nanodot multilayers fabricated by molecular beam epitaxy. b, Bright-field TEM image of a sample with tSi =12nm. The dark areas correspond to the Ge layers. The inset shows a high-resolution TEM of a nanodot. c, AFM image of a single Ge/Si(001) dot layer before overgrowth with Si. d, AFM image of the topmost layer of a sample with tSi =3nm.

the signal can be carried out. The most important artefacts are: residual pump signal on the photodetector, misalignment of the pump and the probe beams, and spot size change as a function of the delay line position. All of these artefacts induce systematic errors in the identification procedures used to extract thermal and acoustic materials properties22–24; they are inherent to this kind of configuration and affect the uncertainty on the identified thermal properties. To fix all of these sources of artefacts, we eliminated any mechanical translation stage and modulation using two heterodyne pump and probe laser beams at slightly different repetition rates. Then there was no need for the lock-in amplifier and the signal was acquiredbyusingonlyanoscilloscope(seetheMethodssection).

HPTR measurements were carried out at the University of

Bordeaux on the samples with 5 and 11Ge layers, and independent 3ω measurementswerecarriedoutatIFWDresdenonsampleswith 11Ge layers, yielding results consistent with those from HPTR. The results are shown in Fig. 2, where the cross-plane thermal resistance per interface R (Fig. 2a) and the thermal conductivity of all of the samples (Fig. 2b) are plotted as a function of multilayer period thickness L. (L is estimated as the sum of tSi and the average amount of Ge per dot layer tGe.) The quoted error bars take into account different sources of uncertainties: for the 3ω measurements they represent confidence intervals (at 68% confidence level) estimated by propagating the uncertainties in the experimental parameters (metal strip widths, electric power and so on) by the Monte Carlo method; for the HPTR the main source of uncertainty is induced by the confidence on fixed parameter values such as the heat capacity of the metal film covering the sample. These values were taken from the literature or obtained from other types of measurement (AFM or profilometry for thicknesses for example) (for details see Supplementary Information). It was previously shown that the 3ω and thermoreflectance measurements compare well to each other25,as confirmed bythe excellentagreement betweenthe results obtained on the samples with 1 Ge layers. The differences between the results obtained on the two different sample sets by HPTR may be ascribed to the slightly different amounts of Ge used in the growth of the two sample sets.

It is illuminating to normalize the thermal resistance values of Fig. 2a by the average amount of Ge contained in each interface, given in terms of its thickness tGe. When this is done, one obtains a nearlyconstantthermalresistivity,ofabout4.5mKW−1 forasingle layer, independent of the period (Fig. 2c). This strongly suggests that transport through the Si regions is ballistic, and resistance is produced by the independent nanodot layers: each layer acts as


3 1 Ge layers HPTR 1 Ge layers

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¬9 m

2 K W

(m K W

(W m

–1 K ρ κ (W m

–1 K cd ω

3 1 Ge layersω3 1 Ge layersω

3 1 Ge layersω

Figure 2 | Thermal response of the Ge nanodot multilayers. a, Experimental thermal resistance divided by the number of layers, for the 5 and 1 layer systems, measured by HPTR, and the 1 layer system measured by the 3ω technique, as a function of the average distance between layers, L. b, Experimental thermal conductivities corresponding to the systems in a. The solid line is the result of the DMM. c, Thermal resistance per barrier, normalized by the average amount of Ge in the barrier (given as an effective length), for the systems in a. d, Thermal conductivities computed by AGFs for four barriers. Diamonds: nanodot barriers. Triangles: flat barriers. The experimental results from b are also shown for comparison.

an individual barrier and the total thermal resistance is the sum of the individual barrier resistances. Such a picture is consistent with the fact that the average phonon mean free path in Si is larger than a hundred nanometres26. The phonon mean free path is thus determined by scattering with the SiGe nanodots, which are arranged in individual layers, perpendicular to the direction of heat propagation, and separated by a distance L between each consecutive layer. In an overly simplistic view, a fully diffusive barrier will have equal transmission and reflection probabilities of 1/2, yielding the limit λ ∼ L for the mean free path. The thermal conductivity can now be evaluated as an integral over frequencies27, obtaining the DMM limit shown by the solid line in Fig. 2b. The measured thermal resistance associated with one individual interface is around 2.5–4 × 10−9 m2 KW−1. This is close to the DMM value in the totally diffuse case (see Fig. 2b), and it is 2–3 times larger than the values reported in refs 6,7.

The above implies that a very precise control over the thermal conductivity value of the nanostructured material can be achieved by varying the period length. As a result of the highly diffusive character of the interfaces, we are able to reach the very low thermal conductivity value of (0.9 ± 0.1)Wm−1 K−1 when using periods of ∼3.7nm (5nanodot layers separated by 4 Si spacings, with a total thickness of 15.5nm and an average barrier resistance of 3.5×10−9 m2 KW−1 in this case). This thermal conductivity is the lowest reported so far for bulk-like Si or SiGe samples, and it is well below the amorphous Si limit of 2.5Wm−1 K−1 (ref. 28). A smaller thermal conductivity of 0.76Wm−1 K−1 was reported only for 10-nm-wide Si nanowires29,30. A value of 1.2Wm−1 K−1 was independently reported on rough Si nanowires30,31.

To rule out the presence of extended defects, we have carried out AFM investigations on large areas of the samples in addition to

TEM. It is in fact known that the occurrence of dislocations during the fabrication of the multilayer will locally disrupt the growth of Si or Ge leading to the formation of either large Ge islands or pits/mounds in the Si cap layer. For the samples with the smallest period and the smallest cross-plane thermal conductivity (tSi = 3 and 6nm) we were not able to detect any such features in an area as large as 60×60µm2 (see Supplementary Information). Thus, we can consider the system as dislocation free, and the low measured thermal conductivities are the result of phonon scattering by the nanostructures alone.

The additive character of the individual interface thermal resistances allows us to engineer regions with accurately defined values of κ, with good spatial resolution down to the 10nm level. In our samples, the shortest measured region was ∼15nm thick, consisting of five interfaces. Previous publications have shown that the thermal conductivity may exhibit a minimum as a function of period length if the periods are reduced below 4nm (for BiSbTe3 superlattices) or 7 nm (for SiGe superlattices; refs 32–34). However, wedidnotobserveanysuchminimumdowntothe∼3.7nmperiod size. As such ‘thermal conductivity minimum’ is related to the onset of wave interference effects across the interfaces, its absence from our results reinforces the conclusion that scattering at the nanodot layers is highly diffusive.

It is intriguing that previous measurements on ‘flat’ SiGe/Si superlattices had reported thermal conductivity values about three times larger than our results and the DMM at room temperature. To elucidate the reasons for this difference, we have carried out an AGF calculation (see the Methods section) of the thermal conductivities of both a flat multilayer and a quantum dot multilayer system, comprising four barriers (Fig. 2d.)

We first computed the thermal conductivity of a bulk Si0.5Ge0.5


Ti:sapphire laser 100 fs, 80 MHz, λ1


Pump beam Probe beam

SampleMicroscope objective

Beam splitter

Signal photodiode

Synchro photodiode

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Ti:sapphire laser 100 fs, 80 MHz + ΔF,

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