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Semiconductor membranes find their widespread use in various research fields targeting medical, biological, environmental, and optical applications. Often such membranes derive their functionality from an inherent nanopatterning, which renders the determination of their, e.g., optical, electronic, mechanical, and thermal properties a challenging task. In this work we demonstrate the non-invasive, all-optical thermal characterization of around 800-nm-thick and 150-$\mu$m-wide membranes that consist of wurtzite GaN and a stack of In$_{0.15}$Ga$_{0.85}$N quantum wells as a built-in light source. Due to their application in photonics such membranes are bright light emitters, which challenges their non-invasive thermal characterization by only optical means. As a solution, we combine two-laser Raman thermometry with (time-resolved) photoluminescence measurements to extract the in-plane (i.e., $c$-plane) thermal conductivity $\kappa_{\text{in-plane}}$ of our membranes. Based on this approach, we can disentangle the entire laser-induced power balance during our thermal analysis, meaning that all fractions of reflected, scattered, transmitted, and reemitted light are considered. As a result of our thermal imaging via Raman spectroscopy, we obtain $\kappa_{\text{in-plane}}\,=\,165^{+16}_{-14}\,$Wm$^{-1}$K$^{-1}$ for our best membrane, which compares well to our simulations yielding $\kappa_{\text{in-plane}}\,=\,177\,$Wm$^{-1}$K$^{-1}$ based on an ab initio solution of the linearized phonon Boltzmann transport equation. Our work presents a promising pathway towards thermal imaging at cryogenic temperatures, e.g., when aiming to elucidate experimentally different phonon transport regimes via the recording of non-Fourier temperature distributions., Comment: Main text (4 figures and 15 pages) and Supplemental Material (3 supplemental figures and 4 pages)

Predicting nanoscale thermal transport in dielectrics requires models, such as the Boltzmann transport equation (BTE), that account for phonon boundary scattering in structures with complex geometries. Although the BTE has been validated against several key experiments, its computational expense limits its applicability. Here, we demonstrate the use of an analytic reduced-order model for predicting the thermal conductivity in dimensionally confined materials, i.e., monolithic and porous thin films, and rectangular and cylindrical nanowires. The approach uses the recently developed "Ballistic Correction Model" (BCM) which accounts for materials' full distribution of phonon mean-free-paths. The model is validated against BTE simulations for a selection of base materials, obtaining excellent agreement. By furnishing a precise yet easy-to-use prediction of thermal transport in nanostructures, our work strives to accelerate the identification of materials for energy-conversion and thermal-management applications., Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in S. A. Hosseini et al., Appl. Phys. Lett. 122 (26): 262202 (2023) and may be found at https://doi.org/10.1063/5.0149792

We present the simultaneous optical and thermal analysis of a freestanding photonic semiconductor membrane made from wurtzite III-nitride material. By linking micro-photoluminescence ($\mu$PL) spectroscopy with Raman thermometry, we demonstrate how a robust value for the thermal conductivity $\kappa$ can be obtained using only optical, non-invasive means. For this, we consider the balance of different contributions to thermal transport given by, e.g., excitons, charge carriers, and heat carrying phonons. Further complication is given by the fact that this membrane is made from direct bandgap semiconductors, designed to emit light based on an In$_{x}$Ga$_{1-x}$N ($x=0.15$) quantum well embedded in GaN. To meet these challenges, we designed a novel experimental setup that enables the necessary optical and thermal characterizations in parallel. We perform micro-Raman thermometry, either based on a heating laser that acts as a probe laser (1-laser Raman thermometry), or based on two lasers, providing the heating and the temperature probe separately (2-laser Raman thermometry). For the latter technique, we obtain temperature maps over tens of micrometers with a spatial resolution less than $1\,\mu\text{m}$, yielding $\kappa\,=\,95^{+11}_{-7}\,\frac{\text{W}}{\text{m}\cdot \text{K}}$ for the $\textit{c}$-plane of our $\approx\,250\text{-nm}$-thick membrane at around room temperature, which compares well to our $\textit{ab initio}$ calculations applied to a simplified structure. Based on these calculations, we explain the particular relevance of the temperature probe volume, as quasi-ballistic transport of heat-carrying phonons occurs on length scales beyond the penetration depths of the heating laser and even its focus spot radius. The present work represents a significant step towards non-invasive, highly spatially resolved, and still quantitative thermometry performed on a photonic membrane., Comment: 28 pages, 14 figures, and Supplemental Material

Deep neural operators can learn operators mapping between infinite-dimensional function spaces via deep neural networks and have become an emerging paradigm of scientific machine learning. However, training neural operators usually requires a large amount of high-fidelity data, which is often difficult to obtain in real engineering problems. Here, we address this challenge by using multifidelity learning, i.e., learning from multifidelity datasets. We develop a multifidelity neural operator based on a deep operator network (DeepONet). A multifidelity DeepONet includes two standard DeepONets coupled by residual learning and input augmentation. Multifidelity DeepONet significantly reduces the required amount of high-fidelity data and achieves one order of magnitude smaller error when using the same amount of high-fidelity data. We apply a multifidelity DeepONet to learn the phonon Boltzmann transport equation (BTE), a framework to compute nanoscale heat transport. By combining a trained multifidelity DeepONet with genetic algorithm or topology optimization, we demonstrate a fast solver for the inverse design of BTE problems.

Nanostructured semiconducting alloys obtain ultra-low thermal conductivity as a result of the scattering of phonons with a wide range of mean-free-paths (MFPs). In these materials, long-MFP phonons are scattered at the nanoscale boundaries whereas short-MFP high-frequency phonons are impeded by disordered point defects introduced by alloying. While this trend has been validated by simplified analytical and numerical methods, an ab-initio space-resolved approach remains elusive. To fill this gap, we calculate the thermal conductivity reduction in porous alloys by solving the mode-resolved Boltzmann transport equation for phonons using the finite-volume approach. We analyze different alloys, length-scales, concentrations, and temperatures, obtaining a very large reduction in the thermal conductivity over the entire configuration space. For example, a ~97% reduction is found for Al$_{0.8}$In$_{0.2}$As with 25% porosity. Furthermore, we employ these simulations to validate our recently introduced "Ballistic Correction Model" (BCM), an approach that estimates the effective thermal conductivity using the characteristic MFP of the bulk alloy and the length-scale of the material. The BCM is then used to provide guiding principles in designing alloy-based nanostructures. Notably, it elucidates how porous alloys such as Si$_{x}$Ge$_{1-x}$ obtain larger thermal conductivity reduction compared to porous Si or Ge, while also explaining why we should not expect similar behavior in alloys such as Al$_{x}$In$_{1-x}$As. By taking into account the synergy from scattering at different scales, we provide a route for the design of materials with ultra-low thermal conductivity.

We introduce a methodology for density-based topology optimization of non-Fourier thermal transport in nanostructures, based upon adjoint-based sensitivity analysis of the phonon Boltzmann transport equation (BTE) and a novel material interpolation technique, the "transmission interpolation model" (TIM). The key challenge in BTE optimization is handling the interplay between real- and momentum-resolved material properties. By parameterizing the material density with an interfacial transmission coefficient, TIM is able to recover the hard-wall and no-interface limits, while guaranteeing a smooth transition between void and solid regions. We first use our approach to tailor the effective thermal-conductivity tensor of a periodic nanomaterial; then, we maximize classical phonon size effects under constrained diffusive transport, identifying a promising new thermoelectric material design. Our method enables the systematic optimization of materials for heat management and conversion and, more broadly, the design of devices where diffusive transport is not valid.

Controlling heat flow at the nanoscales is pivotal to several applications, including thermal energy harvesting and heat management. However, engineering nanostructures is challenging because phonon-boundary interaction, not contemplated by Fourier's law, must be taken into account. Nondiffusive models, such as the Boltzmann transport equation (BTE), have been successfully employed to capture size effects in complex structures; however, their widespread has been hindered by the limited offer of open-source solvers in this space. We fill this void by introducing OpenBTE, an efficient solver for the steady-state phonon BTE in multidimensional structures. This tool is interfaced to first-principles calculations, thus it unlocks the calculations of thermal-related properties with no fitting-parameters. As an example, we employ OpenBTE to compute the temperature and flux maps, as well as the mode-resolved, effective thermal conductivity of Si membranes with infinite and finite thickness. By unlocking fast nanoscale heat transport simulations, OpenBTE may help accelerate the design of nanomaterials for thermal energy applications.

Nanostructured materials enable high thermal transport tunability, holding promises for thermal management and heat harvesting applications. Predicting the effect that nanostructuring has on thermal conductivity requires models, such as the Boltzmann transport equation (BTE), that capture the non-diffusive transport of phonons. Although the BTE has been well validated against several key experiments, notably those on nanoporous materials, its applicability is computationally expensive. Several effective model theories have been put forward to estimate the effective thermal conductivity; however, most of them are either based on simple geometries, e.g., thin films, or simplified material descriptions such as the gray-model approximation. To fill this gap, we propose a model that takes into account the whole mean-free-path (MFP) distribution as well as the complexity of the material's boundaries in infinitely thick films with extruded porosity using uniparameter logistic regression. We validate our approach, which is called the "Ballistic Correction Model" (BCM), against full BTE simulations of a selection of three base materials (GaAs, InAs, and Si) with nanoscale porosity, obtaining excellent agreement. While the key parameters of our method, associated with the geometry of the bulk material, are obtained from the BTE, they can be decoupled and used in arbitrary combinations and scales. We tabulated these parameters for a few cases, enabling the exploration of systems that are beyond those considered in this work. Providing a simple yet accurate estimation of thermal transport in nanostructures, our work sets out to accelerate the discovery of materials for thermal-related applications.

First-principles calculations of thermal transport in homogeneous materials have reached remarkable predicting power. Modeling deterministically phonon transport in nanostructures, however, poses novel challenges; notably, it entails solving as many algebraic equations as the number of combinations of wave vectors in the discretized Brillouen and polarizations. We show that, within the relaxation time approximation of the Boltzmann transport equation (BTE), this issue is resolved by interpolating the phonon distributions in the vectorial phonon mean free paths (MFP) space. The coupling between structure and mode-resolved heat transport is investigated in terms of angular-resolved bulk thermal conductivity and phonon suppression function, the latter being associated primarily to the material's geometry. Our method, termed the anisotropic MFP-BTE (aMFP-BTE), allows for fast and accurate thermal conductivity calculations in nanomaterials regardless of the number of phonon branches and wave vectors. Furthermore, it naturally blends with first-principles thermal transport calculations, therefore allowing for multiscale, parameter-free simulations. We apply the aMFP-BTE to compute the mode-resolved effective thermal conductivity of porous Si membranes, achieving up to 50x speed with respect to the case with no interpolation. The proposed approach unlocks the engineering of novel nanostructures, with applications to thermoelectrics and heat management.

We introduce dPV, an end-to-end differentiable photovoltaic (PV) cell simulator based on the drift-diffusion model and Beer-Lambert law for optical absorption. dPV is programmed in Python using JAX, an automatic differentiation (AD) library for scientific computing. Using AD coupled with the implicit function theorem, dPV computes the power conversion efficiency (PCE) of an input PV design as well as the derivative of the PCE with respect to any input parameters, all within comparable time of solving the forward problem. We show an example of perovskite solar-cell optimization and multi-parameter discovery, and compare results with random search and finite differences. The simulator can be integrated with optimization algorithms and neural networks, opening up possibilities for data-efficient optimization and parameter discovery.

The addition of porosity to thermoelectric materials can significantly increase the figure of merit, ZT, by reducing the thermal conductivity. Unfortunately, porosity is also detrimental to the thermoelectric power factor in the numerator of the figure of merit ZT. In this manuscript we derive strategies to recoup electrical performance in nanoporous Si by fine tuning the carrier concentration and through judicious design of the pore size and shape so as to provide energy selective electron filtering. In this study, we considered phosphorus doped silicon containing discrete pores that are either spheres, cylinders, cubes, or triangular prisms. The effects from these pores are compared with those from extended pores with circular, square and triangular cross sectional shape, and infinite length perpendicular to the electrical current. A semiclassical Boltzmann transport equation is used to model Si thermoelectric power factor. This model reveals three key results: The largest enhancement in Seebeck coefficient occurs with cubic pores. The fractional improvement is about 15% at low carrier concentration ($< 10^{20}\ \mathrm{1/cm^3}$) up to 60% at high carrier population with characteristic length around $\sim 1\ \mathrm{nm}$. To obtain the best energy filtering effect at room temperature, nanoporous Si needs to be doped to higher carrier concentration than is optimal for bulk Si. Finally, in $n$-type Si thermoelectrics the electron filtering effect that can be generated with nanoscale porosity is significantly lower than the ideal filtering effect; nevertheless, the enhancement in the Seebeck coefficient that can be obtained is large enough to offset the reduction in electrical conductivity caused by porosity.

Self-affine morphology of random interfaces governs their functionalities across tribological, geological, (opto-)electrical and biological applications. However, the knowledge of how energy carriers or generally classical/quantum waves interact with structural irregularity is still incomplete. In this work, we study vibrational energy transport through random interfaces exhibiting different correlation functions on the two-dimensional hexagonal lattice. We show that random interfaces at the atomic scale are Cantor composites populated on geometrical fractals, thus multifractals, and calculate their quantized conductance using atomistic approaches. We obtain a universal scaling law, which contains self-similarity for mass perturbation, and exponential scaling of structural irregularity quantified by fractal dimension. The multifractal nature and Cantor-composite picture may also be extendable to charge and photon transport across random interfaces.

Phonon size effects induce ballistic transport in nanomaterials, challenging Fourier's law. Nondiffusive heat transport is captured by the Peierls-Boltzmann transport equation (BTE), commonly solved under the relaxation time approximation (RTA), which assumes diagonal scattering operator. Although the RTA is accurate for many relevant materials over a wide range of temperatures, such as silicon, it underpredicts thermal transport in most two-dimensional (2D) systems, notoriously graphene. Here we present a formalism, based on the BTE with the full collision matrix, for computing the effective thermal conductivity of arbitrarily patterned 2D materials. We apply our approach to porous graphene and find strong heat transport suppression in configurations with feature sizes of the order of micrometers; this result, which is rooted in the large generalized phonon MFPs in graphene, corroborates the possibility of strong thermal transport tunability by relatively coarse patterning. Lastly, we present a promising material configuration with low thermal conductivity. Our method enables the parameter-free design of 2D materials for thermoelectric and thermal routing applications.

In this study, we use the transient thermal grating optical technique \textemdash a non-contact, laser-based thermal metrology technique with intrinsically high accuracy \textemdash to investigate room-temperature phonon-mediated thermal transport in two nanoporous holey silicon membranes with limiting dimensions of 100 nm and 250 nm respectively. We compare the experimental results to ab initio calculations of phonon-mediated thermal transport according to the phonon Boltzmann transport equation (BTE) using two different computational techniques. We find that the calculations conducted within the Casimir framework, i.e. based on the BTE with the bulk phonon dispersion and diffuse scattering from surfaces, are in quantitative agreement with the experimental data, and thus conclude that this framework is adequate for describing phonon-mediated thermal transport through holey silicon membranes with feature sizes on the order of 100 nm.

X-ray diffraction (XRD) data acquisition and analysis is among the most time-consuming steps in the development cycle of novel thin-film materials. We propose a machine-learning-enabled approach to predict crystallographic dimensionality and space group from a limited number of thin-film XRD patterns. We overcome the scarce-data problem intrinsic to novel materials development by coupling a supervised machine learning approach with a model agnostic, physics-informed data augmentation strategy using simulated data from the Inorganic Crystal Structure Database (ICSD) and experimental data. As a test case, 115 thin-film metal halides spanning 3 dimensionalities and 7 space-groups are synthesized and classified. After testing various algorithms, we develop and implement an all convolutional neural network, with cross validated accuracies for dimensionality and space-group classification of 93% and 89%, respectively. We propose average class activation maps, computed from a global average pooling layer, to allow high model interpretability by human experimentalists, elucidating the root causes of misclassification. Finally, we systematically evaluate the maximum XRD pattern step size (data acquisition rate) before loss of predictive accuracy occurs, and determine it to be 0.16{\deg}, which enables an XRD pattern to be obtained and classified in 5.5 minutes or less., Comment: Accepted with minor revisions in npj Computational Materials, Presented in NIPS 2018 Workshop: Machine Learning for Molecules and Materials

Bayesian inference is a widely used and powerful analytical technique in fields such as astronomy and particle physics but has historically been underutilized in some other disciplines including semiconductor devices. In this work, we introduce Bayesim, a Python package that utilizes adaptive grid sampling to efficiently generate a probability distribution over multiple input parameters to a forward model using a collection of experimental measurements. We discuss the implementation choices made in the code, showcase two examples in photovoltaics, and discuss general prerequisites for the approach to apply to other systems.

Achieving low thermal conductivity and good electrical properties is a crucial condition for thermal energy harvesting materials. Nanostructuring offers a very powerful tool to address both requirements: in nanostructured materials, boundaries preferentially scatter phonons compared to electrons. The search for low-thermal-conductivity nanostructures is typically limited to materials with simple crystal structures, such as silicon, because of the complexity arising from modeling branch- and wave vector- dependent nanoscale heat transport. Using the phonon mean-free-path (MFP) dependent Boltzmann transport equation, a model that overcomes this limitation, we compute thermal transport in 75 nanoporous half-Heusler compounds for different pore sizes. We demonstrate that the optimization of thermal transport in nanostructures should take into account both bulk thermal properties and geometry-dependent phonon suppression, two aspects that are typically engineered separately. In fact, our work predicts that, given a set of bulk materials and a system geometry, the ordering of the thermal conductivity of the nanostructure does not necessarily align with that of the bulk: We show that what dictates thermal transport is the interplay between the bulk MFP distribution and the nanostructuring length scale of the material. Finally, we derive a thermal transport model that enables fast systems screening within large bulk material repositories and a given geometry. Our study motivates the need for a holistic approach to engineering thermal transport and provides a method for high-throughput materials discovery.

Nanostructured semiconducting materials are promising candidates for thermoelectrics due to their potential to suppress phonon transport while preserving electrical properties. Modeling phonon-boundary scattering in complex geometries is crucial for predicting materials with high conversion efficiency. However, the simultaneous presence of ballistic and diffusive phonons challenges the development of models that are both accurate and computationally tractable. Using the recently developed first-principles Boltzmann transport equation (BTE) approach, we investigate diffusive phonons in nanomaterials with wide mean-free-path (MFP) distributions. First, we derive the short MFP limit of the suppression function, showing that it does not necessarily recover the value predicted by standard diffusive transport, challenging previous assumptions. Second, we identify a Robin type boundary condition describing diffuse surfaces within Fourier's law, extending the validity of diffusive heat transport in terms of Knudsen numbers. Finally, we use this result to develop a hybrid Fourier/BTE approach to model realistic materials, obtaining excellent agreement with experiments. These results provide insight on thermal transport in materials that are within experimental reach and open opportunities for large-scale screening of nanostructured thermoelectric materials.

Nanoporous materials are a promising platform for thermoelectrics in that they offer high thermal conductivity tunability while preserving good electrical properties, a crucial requirement for high- effciency thermal energy conversion. Understanding the impact of the pore arrangement on thermal transport is pivotal to engineering realistic materials, where pore disorder is unavoidable. Although there has been considerable progress in modeling thermal size effects in nanostructures, it has remained a challenge to screen such materials over a large phase space due to the slow simulation time required for accurate results. We use density functional theory in connection with the Boltzmann transport equation, to perform calculations of thermal conductivity in disordered porous materials. By leveraging graph theory and regressive analysis, we identify the set of pores representing the phonon bottleneck and obtain a descriptor for thermal transport, based on the sum of the pore-pore distances between such pores. This approach provides a simple tool to estimate phonon suppression in realistic porous materials for thermoelectric applications and enhance our understanding of heat transport in disordered materials.

While thermal anisotropicity is a desirable materials property for many applications, including transverse thermoelectrics and thermal management in electronic devices, it remains elusive in practical natural compounds. In this work, we show how nanoporous materials with anisotropic pore lattices can be used as a platform for inducing strong heat transport directionality in isotropic materials. Using density functional theory and the phonon Boltzmann transport equation, we calculate the phonon-size effects and thermal conductivity of nanoporous silicon with different anisotropicpore lattices. Our calculations predict a strong directionality in the thermal conductivity, dictated by the difference in the pore-pore distances along the two Cartesian axes. As the space between pores along the direction of the applied temperature gradient represents the phonon bottleneck, an anisotropic pores lattice distortion induces directionality in heat transport. Using Fourier's law, we also compute diffusive heat transport for the same geometries obtaining significantly smaller anisotropicity, revealing the crucial role of phonon-size effects in tuning thermal transport directionality. Besides enhancing our understanding of nanoscale heat transport, our results demonstrate the promise of nanoporous materials for modulating anisotropy in thermal conductivity.

Boundary-engineering in nanostructures has the potential to dramatically impact the development of materials for high-efficiency conversion of thermal energy directly into electricity. In particular, nanostructuring of semiconductors can lead to strong suppression of heat transport with little degradation of electrical conductivity. Although this combination of material properties is promising for thermoelectric materials, it remains largely unexplored. In this work, we introduce a novel concept, the directional phonon suppression function, to unravel boundary-dominated heat transport in unprecedented detail. Using a combination of density functional theory and the Boltzmann transport equation, we compute this quantity for nanoporous silicon materials. We first compute the thermal conductivity for the case with aligned circular pores, confirming a significant thermal transport degradation with respect to the bulk. Then, by analyzing the information on the directionality of phonon suppression in this system, we identify a new structure of rectangular pores with the same porosity that enables a four-fold decrease in thermal transport with respect to the circular pores. Our results illustrate the utility of the directional phonon suppression function, enabling new avenues for systematic thermal conductivity minimization and potentially accelerating the engineering of next-generation thermoelectric devices.

Electronic devices composed of single molecules constitute the ultimate limit in the continued downscaling of electronic components. A key challenge for single-molecule electronics is to control the temperature of these junctions. Controlling heating and cooling effects in individual vibrational modes, can in principle, be utilized to increase stability of single-molecule junctions under bias, to pump energy into particular vibrational modes to perform current-induced reactions or to increase the resolution in inelastic electron tunneling spectroscopy by controlling the life-times of phonons in a molecule by suppressing absorption and external dissipation processes. Under bias the current and the molecule exchange energy, which typically results in heating of the molecule. However, the opposite process is also possible, where energy is extracted from the molecule by the tunneling current. Designing a molecular 'heat sink' where a particular vibrational mode funnels heat out of the molecule and into the leads would be very desirable. It is even possible to imagine how the vibrational energy of the other vibrational modes could be funneled into the 'cooling mode', given the right molecular design. Previous efforts to understand heating and cooling mechanisms in single molecule junctions, have primarily been concerned with small models, where it is unclear which molecular systems they correspond to. In this paper, our focus is on suppressing heating and obtaining current-induced cooling in certain vibrational modes. Strategies for cooling vibrational modes in single-molecule junctions are presented, together with atomistic calculations based on those strategies. Cooling and reduced heating are observed for two different cooling schemes in calculations of atomistic single-molecule junctions., Comment: 18 pages, 6 figures

In this book chapter we provide the definition of "Simulating Nanoscale Heat Transport", broadly regarded as modeling heat conduction beyond Fourier's law. We primarely focus on incoherent transport, which is dominated by scattering between phonons and the boundaries of a given material. After introducing the Boltzmann Transport Equation (BTE) for phonons, we elaborate on different numerical techniques for solving nanoscale heat transport includind deterministic and Monte Carlo based method. A discussion on the need for multiscale modeling concludes the chapter.

Nanostructured materials exhibit low thermal conductivity because of the additional scattering due to phonon-boundary interactions. As these interactions are highly sensitive to the mean free path (MFP) of a given phonon mode, MFP distributions in nanostructures can be dramatically distorted relative to bulk. Here we calculate the MFP distribution in periodic nanoporous Si for different temperatures, using the recently developed MFP-dependent Boltzmann Transport Equation. After analyzing the relative contribution of each phonon branch to thermal transport in nanoporous Si, we find that at room temperature optical phonons contribute 18 % to heat transport, compared to 5% in bulk Si. Interestingly, we observe a steady thermal conductivity in the nanoporous materials over a temperature range 200 K < T < 300 K, which we attribute to the ballistic transport of acoustic phonons with long intrinsic MFP. These results, which are also consistent with a recent experimental study, shed light on the origin of the reduction of thermal conductivity in nanostructured materials, and could contribute to multiscale heat transport engineering, in which the bulk material and geometry are optimized concurrently.

Tuning thermal transport in nanostructured materials is a powerful approach to develop high-efficiency thermoelectric materials. Using a recently developed approach based on the phonon mean free path dependent Boltzmann transport equation, we compute the effective thermal conductivity of nanoporous materials with pores of various shapes and arrangements. We assess the importance of pore-pore distance in suppressing thermal transport, and identify the pore arrangement that minimizes the thermal conductivity, composed of a periodic arrangement of two misaligned rows of triangular pores. Such a configuration yields a reduction in the thermal conductivity of more than $60 \%$ with respect the simple circular aligned case with the same porosity., Comment: 4 pages, 4 figures, 1 table

We develop a computational framework, based on the Boltzmann transport equation, with the ability to compute the thermal transport in nanostructured materials of any geometry using as the only input the bulk thermal conductivity accumulation function. The main advantage of our method is twofold. First, while the scattering times and dispersion curves are unknown for most materials, the phonon mean free path distribution can be directly obtained by experiments. As a consequence, a wider range of materials can be simulated than with a frequency-dependent approach. Second, when phonon dispersions are available from first principles calculations, our approach allows one to include easily the whole Brillouen zone in the calculations without discretizing the phonon frequencies for all polarizations, reducing considerably the computational effort. Furthermore, after deriving the ballistic and diffusive limits of our model, we develop a multi-scale scheme that couples phonon transport across different scales, enabling efficient simulations of materials with wide phonon mean free path distributions. After we validate our model against the frequency-dependent approach, we apply the method to porous silicon membranes and find good agreement with experiments on mesoscale pores. By enabling the investigation of thermal transport in unexplored nanostructured materials, our method has potential to advance high-efficiency thermoelectric devices.