Fractal Lé vy Heat Transport in Nanoparticle Embedded Semiconductor Alloys Amr M.S. Mohammed, † Yee Rui Koh, † Bjorn Vermeersch, † Hong Lu, ‡ Peter G. Burke, ‡ Arthur C. Gossard, ‡ and Ali Shakouri ,† † Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, United States ‡ Materials Department, University of California, Santa Barbara, California 93106, United States * S Supporting Information ABSTRACT: Materials with embedded nanoparticles are of consid- erable interest for thermoelectric applications. Here, we experimen- tally characterize the effect of nanoparticles on the recently discovered Le ́ vy phonon transport in semiconductor alloys. The fractal space dimension α ≈ 1.55 of quasiballistic (superdiffusive) heat conduction in (ErAs) x :InGaAlAs is virtually independent of the Er content 0.001 < x < 0.1 but instead controlled by alloy scattering of the host matrix. The increased nanoparticle concentration does reduce the diffusive recovery length by an order of magnitude. The bulk conductivity drops by 3-fold, in close agreement with a Callaway model. Our results may provide helpful hints toward engineering superdiffusive heat transport similar to what has been achieved with light in Le ́ vy glasses. KEYWORDS: Nanoparticles, Ballistic heat transport, Superdiffusion, Le ́ vy flight, Thermal Conductivity, Thermoelectric materials Q uasiballistic thermal transport, that is, transport occurring over length scales comparable to phonon mean free paths (MFPs), deviates significantly from diffusive predictions. 1−4 As a result, heat conduction in nanostructures can be very different from that in bulk materials. A detailed understanding of the underlying transport processes is crucial for many technological applications including electronic devices 5,6 and thermoelectric (TE) materials. The performance of the latter is governed by a dimensionless figure of merit ZT = σS 2 T/k. An ideal TE material therefore must have a large Seebeck coefficient S and should act as a crystal for electrons (large electrical conductivity σ) but as a glass for phonons (low thermal conductivity k). 7 Due to inherent trade-offs between S and σ, most TE optimization efforts have focused on lowering the thermal conductivity beyond the alloy limit. The introduction of heavy point defects 8 and nanostructuring 9−14 have proven both theoretically and experimentally to be a successful approach in ZT enhancement. Following the report that semimetallic nanoparticles (NPs) can beat the thermal conductivity alloy limit by 2-fold, 15 a series of studies have investigated the impact of NPs on TE material performance. 16−19 As NP scattering targets a limited portion of the MFP spectrum, additional improvements have been proposed by introducing scatterers at all relevant length scales. 20 ZTs exceeding 2 over the 900−1000K temperature range were reported in nanostructured PbTe and distorted rock-salt SnSe. 20,21 The ultralow thermal conductivity of the latter was attributed to phonon dispersion anisotropy and anharmonicity. Such findings indicate the importance and practical relevance of gaining knowledge about phonon population properties and fundamental heat conduction mechanisms for TE optimization. Ab initio Boltzmann transport equation (BTE) analyses have recently shown that in semiconductor alloys the Fourier theory breaks down in a very particular way. 22 Namely, the random motion of quasiballistic thermal energy is no longer Brownian but instead governed by so-called Lé vy transport. This fractional superdiffusion process induces characteristic random walk patterns with fractal spatial dimension 1 < α < 2. 23 The anomalous behavior gradually recovers to regular diffusion over a characteristic length scale u BD . Both Le ́ vy parameters can be determined experimentally in conjunction with the bulk thermal conductivity using laser thermoreflectance and provide valuable microscopic metrics about the phonon population. 24 Given the relevance for TE applications and aforementioned prevalence of nanostructuring in this context, the question arises how the inclusion of nanoparticles inside an alloy host matrix affects the Lé vy characteristics of the thermal transport. In this Letter, we experimentally characterize microscale heat conduction inside InGaAlAs with embedded self-assembling ErAs nanoparticles with a narrow size distribution. Varying the NP concentration from 0.1% to 10%, which is achieved by changing the volumetric percentage of ErAs preserves a near constant fractal dimension α ≈ 1.55, whereas u BD drops from Received: November 20, 2014 Revised: February 1, 2015 Letter pubs.acs.org/NanoLett © XXXX American Chemical Society A DOI: 10.1021/nl5044665 Nano Lett. XXXX, XXX, XXX−XXX