Thermal Diffusion Model Research Articles

Assessing thermal resistance in fusion bond layers of 3D heterogeneous electronics packaging

We investigated thermal management challenges in semiconductor devices that use fusion bonding, with a focus on measuring the thermal resistance of the bonded interface. Starting with two fusion-bonded silicon wafers, we thinned the top wafer to below 100 μm and used frequency domain thermoreflectance to measure the effective thermal boundary resistance of the fusion bond. We established the measurement uncertainty with a multilayer thermal diffusion model, considering the effects of the top Si layer thickness and laser spot size. The thermal boundary resistance was measured to be 0.2 and 0.5 m2K/MW for two types of bonded wafers, with total SiO2 film thicknesses of 200 and 470 nm at the fusion bond layer, respectively. We also created defects at the interface and imaged these defects with the thermal phase response at specific frequencies chosen according to a thermal model. The method described can be applied to other types of wafer-to-wafer or die-to-die bonding including direct, polymeric, and metallic bonds.

An instrumentation guide to measuring thermal conductivity using frequency domain thermoreflectance (FDTR).

Frequency Domain Thermoreflectance (FDTR) is a versatile technique used to measure the thermal properties of thin films, multilayer stacks, and interfaces that govern the performance and thermal management in semiconductor microelectronics. Reliable thermal property measurements at these length scales (≈10nm to ≈10μm), where the physics of thermal transport and phonon scattering at interfaces both grow in complexity, are increasingly relevant as electronic components continue to shrink. While FDTR is a promising technique, FDTR instruments are generally home-built; they can be difficult to construct, align, and maintain, especially for the novice. Our goal here is to provide a practical resource beyond theory that increases the accessibility, replicability, and widespread adoption of FDTR instrumentation. We provide a detailed account of unpublished insights and institutional knowledge that are critical for obtaining accurate and repeatable measurements of thermal properties using FDTR. We discuss component selection and placement, alignment procedures, data collection parameters, common challenges, and our efforts to increase measurement automation. In FDTR, the unknown thermal properties are fit by minimizing the error between the phase lag at each frequency and the multilayer diffusive thermal model solution. For data fitting and uncertainty analysis, we compare common numerical integration methods, and we compare multiple approaches for fitting and uncertainty analysis, including Monte Carlo simulation, to demonstrate their reliability and relative speed. The instrument is validated with substrates of known thermal properties over a wide range of isotropic thermal conductivities, including Borofloat silica, quartz, sapphire, and silicon.

Hybrid multiscale computational approach for heat transfer in GaN semiconductors

Abstract With ongoing advancements in semiconductor technology, understanding thermal transport in semiconductor devices—where phonons are the primary charge carriers—is crucial. This involves a multiscale process in which phenomena at different scales are governed by distinct control equations, necessitating various numerical methods. In this context, this paper introduces a novel multiscale simulation method called the Second-Order Reconstruction Operator Finite Difference Lattice Boltzmann Method (ROsFDLBM). This method effectively combines the Lattice Boltzmann Method (LBM) with the Finite Difference Method (FDM). ROsFDLBM employs reconstruction operators to accurately map the relationship between distribution functions and macroscopic physical quantities, significantly reducing errors caused by the transmission of interface information. The efficacy of this coupling scheme has been confirmed through an interface-coupled thermal diffusion model. Comparative analyses have revealed that the relative error of ROsFDLBM does not exceed 1.2%, demonstrating its superior accuracy. Further simulations of GaN devices have confirmed that traditional macroscopic methods tend to underestimate the temperature in the heat source area at micro-nano scales, with discrepancies reaching up to 60.9 K. Overall, the ROsFDLBM aligns with current research findings and offers an accurate and efficient simulation strategy for addressing complex heat transfer problems.

Study on the UV FEL single-shot damage threshold of an Au thin film.

The damage threshold of an Au-coated flat mirror, one of the reflective optics installed on the FEL-2 beamline of the Dalian Coherent Light Source, China, upon far-UV free-electron laser irradiation is evaluated. The surface of the coating is characterized by profilometer and optical microscope. A theoretical approach of the phenomenon is also presented, by application of conventional single-pulse damage threshold calculations, a one-dimensional thermal diffusion model, as well as finite-element analysis with ANSYS.

An upscaling model for simulation of geothermal processes in stratified formations

In stratified porous media, non-uniform velocity between layers combined with thermal conduction across layers causes spreading of the thermal front: thermal Taylor dispersion. Conventional upscaling not accounting for this heterogeneity within simulation grid blocks underestimates thermal dispersion, leading to overestimation of thermal breakthrough time. We derive a model for effective longitudinal thermal diffusivity in the direction of flow, αeff, to represent the effective thermal dispersion in two-layer media. αeff, accounting for thermal Taylor dispersion, is much greater than the thermal diffusivity of the rock itself. We define a dimensionless number, NTC, a ratio of times for longitudinal convection to transverse conduction, as an indicator of transverse thermal equilibration of the system during cold- or hot-water injection. For NTC > 5, thermal dispersion in the two-layer system closely approximates a single layer with αeff. This suggests a two-layer medium satisfying NTC > 5 can be combined into a single layer with an effective longitudinal thermal diffusivity αeff. In application to a geothermal reservoir, one can apply the model to perform upscaling in stages, i.e. combining two layers satisfying the NTC criterion in each stage. The αeff model accounting for the fine-scale heterogeneity within simulation grid blocks would enhance the prediction accuracy of thermal breakthrough time and thus thermal lifetime.

Multiple interaction electron beam powder bed fusion for controlling melt pool dynamics and improving surface quality

Low surface quality of fabricated parts and long processing times are two commonly stated reasons against a more wide-spread adoption of electron beam powder bed fusion (PBF-EB) as a commercially viable manufacturing technique. Aside from the large particle size distribution (45–150 µm) of processed powders, the surface roughness, limited structural resolution and process instabilities are attributed to unwanted dynamics of the melt pool, like spattering, balling, evaporation and melt pool displacement. As a result, a contouring step using slow but stable melting parameters is usually added to the process to improve surface quality, leading to increased processing times.In this study, we demonstrate how the surface quality can be improved by supplying the energy for melt pool formation over multiple, timely separated interactions. To investigate the effect of this multiple interaction processing (MIP), series of single line melt structures with increasing number of interactions were fabricated, characterized and corresponding thermal diffusion simulations performed. The experimental results show how MIP allows to lower surface roughness of melted structures through a reduction of spattering, balling and melt pool displacement during processing, while our simulation model identifies these benefits to be caused by a significant reduction in surface temperature and thus evaporation, with increasing number of interactions for melt pool formation. Our model further demonstrates, how a relatively simple, computationally inexpensive and thus scalable thermal diffusion model is capable of providing valuable information about thermal conditions during melt pool formation. Lastly, we introduce a few simple equations to calculate appropriate MIP parameters for any kind of scan strategy.

Evaluation of thermal radiation and Lewis number effects on oscillatory thermal-diffusive instabilities of counterflow premixed flame fed with moisty biomass particles

In this study, a thermal-diffusive model using a sinusoidal function for the oscillation term is presented. The model is employed to evaluate the pulsating behavior in a premixed counterflow flame fed with biomass particle cloud considering Lewis number and thermal radiation effects. In order to scrutinize the pulsating characteristics of a flame, time-related governing equations are written for the assumed multi-zone system and analytically solved utilizing Matlab and Mathematica software. To enforce continuity, suitable matching conditions for the temperature and heat flux at the interfaces are derived. Time-dependent characteristics of burning velocity, flame temperature, flame front position, reactants temperature, and mass distributions are presented, taking into account effective Lewis number, thermal radiation, and strain rate influences. An acceptable comparison with numerical simulations affirms the reliability of the mathematical procedure. The obtained results indicated that the pulsating instability of biomass-fueled premixed flames in counterflow arrangements significantly relevant not only to the effective Lewis number but also to thermal radiation transferred from the reaction zone to unburnt reactants. Furthermore, it is found that in counterflow premixed flames, strain rate, in addition to Lewis number and thermal radiation, can be the key to controlling the individual flame oscillation modes.

Evolutionary equations for the disturbed flame stabilised at the flat burner

The dynamics of the burner-stabilised flame is investigated theoretically in the frame of diffusive-thermal model. The nonlinear system of equations describing dynamics of curved flame front and the flame front temperature is derived in the quasi-stationary approximation from the basic diffusive-thermal model with one-step chemical reaction of Arrhenius type. The reduced model successfully describes nonlinear flame oscillation that is confirmed by comparison of oscillation characteristics obtained in the reduced and full models with the volumetric chemical reaction. The comparison of the instability increments depending on wave numbers agrees with the increments following from the linear stability analysis. The proposed model makes it possible to significantly reduce calculations by reducing the dimension of the system, as well as to obtain estimates of the critical parameters at which oscillations occur. Specifying the critical parameters by simple model will facilitate the performance of complex calculations in the models with detailed kinetics of chemical combustion reactions. A reduced model that describes the flame front spatial instability is proposed too.

DMD analysis on sporadic flame behaviors in low-Lewis-number counterflow under microgravity

In order to deepen the understanding on the quasi-steady flame behaviors in counterflow field involving mixtures with low-Lewis number (Le) near lean limit, the experimental and numerical analyses for sporadic flames were carried out. The experiment was conducted with the mixture of Le 0.3 and 0.5 under microgravity environment using parabolic flights of an airplane to eliminate the effect of buoyancy for low-stretch conditions. Three-dimensional numerical simulation with simplified diffusive thermal model was carried out to investigate the three-dimensional structure of sporadic flames. The experimental observations showed the change in the conditions for sporadic flames with changes in Le. The sporadic flames of Le 0.3 could be observed under leaner conditions and endured higher stretch circumstances compared to those observed with Le 0.5. The numerical simulation also reproduced this trend and proved that Le has an important role in the conditions for sporadic flames. To understand the characteristic structure of sporadic flames, the analysis with dynamic mode decomposition (DMD) was conducted. Through this analysis, three characteristic structure modes—planar, radial, and circular—were identified. The decomposed modes showed that the domain of counterflow field could be divided into two; the area forming sporadic flames near the counterflow axis and the area splitting and transporting flame pieces at outer part of the counterflow field. The characteristic length scale was also analyzed using fast Fourier transform (FFT). FFT analysis quantified the characteristic length scale of the decomposed mode of radial and circular. It was also found that the characteristic length scale in radial and angular direction had the same value.

Thermal diffusion, exhaust gas recirculation and blending effects on lean premixed hydrogen flames

Thermodiffusively-unstable lean premixed hydrogen flames are investigated using two-dimensional direct numerical simulation employing finite-rate chemical kinetics. Three databases are generated focussing on the inclusion of the Soret effect, the recirculation of exhaust gas, and blending with methane. A simple rescaling of a classic thermal diffusion model is presented and shown to mimic multicomponent diffusion with very low computational cost and little-to-no loss in accuracy. It is also shown that a previously developed model for mean local flame speeds in lean premixed hydrogen flames [1] can still be used provided Soret effects are taken into account in one-dimensional calculations. The addition of exhaust gas to the unburned mixture is found to enhance thermodiffusive instability; the primary mechanism for this was shown to be the highly-efficient third-body nature of water, with the reduction of adiabatic flame temperature a second-order effect. Again, the existing mean local flame speed model proved sufficient. Finally, blending with methane was found to reduce the thermodiffusive response of the flame, more so than the existing model suggests, despite adjustment of the fuel Lewis number; an adapted model is presented to account for this.

Modeling 4.3 billion years of water history on Phobos

Phobos, one of the Martian moonlets, has been the focus of long-standing scientific investigation, particularly regarding its origin: whether it was formed by a giant impact or a captured asteroid. In this study, we investigate the long-term internal evolution of Phobos over 4.3 billion years using a 3-dimensional thermal diffusion and water transport model, named ASTRA. The model newly implements reflected visible light and infrared radiation from Mars, orbital evolution of Phobos, water vapor adsorption on rock grain surfaces, and suppressed permeability due to the surface dust layer in addition to the previous study, allowing us to simulate different grain sizes of 1, 10, 100, and 1000 μm, adsorption coefficients of 1 and 10 kg m−3, and initial water contents of 0.1, 1 or 10 wt%.Our simulation results revealed that without the water vapor adsorption effect, the water inside Phobos would be lost over several billion years for an initial water content of 0.1 wt%. However, when water vapor adsorption was considered, scenarios emerge in which Phobos could retain water to the present day. In the case of a grain size of around 100 μm, Phobos could still continuously release water flux of 10−4–10−3 g s−1 for an initial water content of 0.1 wt%, and 10−2–1 g s−1 for an initial water content of >1 wt%. Furthermore, our research shows the possibility of subsurface condensed water ice in deep high latitude regions and the formation of a gas torus by escaping water-related molecules. If the future MMX mission can measure the water-related ions near the gas torus of Phobos with much >104 cm−2 s−1, the origin of Phobos is most likely the captured asteroid with an initial water content of >1%. For further detailed analysis, our results emphasize the importance of exploring surface soil parameters through soil sample return by the MMX mission.

Behavior analysis and threshold prediction of laser-induced damage in fused silica based on 2-D thermal diffusion model

Laser-induced damage is a bottleneck problem that restricts the development of high-power laser facilities. According to the theory of carriers contributed thermal conductivity, we established a 2-D thermal diffusion model for fused silica. This model provides a detailed description of the dynamic evolution of the temperature field distribution during laser pulse loading. The heat deposition core, which consists of the defect and the high-temperature fused silica, continuously absorbs the laser energy. This results in forming a solid absorption wavefront that diffuses in the anti-laser transmission direction. The results of the numerical simulation and the experimental results show that the evolution law of the heat deposition core in the energy deposition process is directly related to the final damage state. According to the evolution characteristics of the instantaneous peak temperature and its position of the heat deposition core, we calculated the critical parameter αth, whose amplitude indicates the maximum absorption capacity of the defects that optical elements can carry. Further numerical simulations established the mathematical correlation of the αth with the laser pulse waveform. The results of the study provide an important theoretical basis for predicting and evaluating the laser-induced damage of optical elements under high-power laser radiation.

Numerical implementation of level set image recovery based on partial differential equation model

Abstract Image recovery is a class of ill-posed inverse problems, so image recovery has been a challenging and fundamental topic in image processing. In this paper, firstly, to recover blurred images with noise, an adaptive fully variable partial differential image recovery model is proposed based on the fully variable partial differential equation model and Gaussian thermal diffusion model, where the adaptiveness is achieved logarithmically by regularizing the edge detection operator. Secondly, a finite difference method, which covers the gradient and scatters operators in the partial differential equation, is applied to the problem of fixing the global range boundary conditions of the image. Finally, the gradient of the image is restored, and then the gradient domain of the image is solved to obtain the complete recovered image. The results show that in the evaluation of the image restoration effect aspect by two indexes, PSNR and SSIM, the value of the adaptive fully variable partial differential image restoration model increases from 28 to 32, and the value of SSIM increases from 0.70 to 0.88, which has good image restoration performance. The adaptive fully variable partial differential image restoration model proposed in this paper has the characteristics of good edge protection and fast convergence, which is of guiding significance for the numerical implementation of image restoration techniques.

Anchored and Lifted Diffusion Flames Supported by Symmetric and Asymmetric Edge Flames

Numerous combustion applications are concerned with the stabilization of diffusion flames formed by injecting gaseous fuels into a co-flowing stream containing an oxidizer. The smooth operation of these devices depends on the attachment and lift-off characteristics of the edge flame at the base of the diffusion flame. In this paper, we address fundamental issues pertinent to the structure and dynamics of edge flames, which have attributes of both premixed and diffusion flames. The adopted configuration is the mixing layer established in the wake of a splitter plate where two streams, one containing fuel and the other oxidizer, merge. The analysis employs a diffusive-thermal model which, although it excludes effects of gas expansion, systematically includes the influences of the overall flow rate, unequal strain rates in the incoming streams, stoichiometry, differential and preferential diffusion, heat loss and gas–solid thermal interaction, and their effect on the edge structure, speed, and temperature. Conditions when the edge flame is anchored to the plate, lifted-off and stabilized in the flow, or blown-off, are identified. Two stable modes of stabilization are observed for lifted flames; the edge flame either remains stationary at a specified location or undergoes spontaneous oscillations along a direction that coincides with the trailing diffusion flame.

Numerical Investigation on Combustion and Heating Characteristics of Metal Fiber Burner

A metal fiber burner is proposed for structural thermal tests for the test model of hypersonic vehicles. In order to guide the design of the burner, we conduct Fluent simulations on a cylindrical metal fiber burner with a properly selected thermal diffusion model, a heat transfer model between the gas and solid phase, a radiation model with a metal porous media, and a gaseous combustion chemistry mechanism. After validating the simulation, the combustion and heating characteristics of the burner were further analyzed. The results show that under the condition of no specimen, when the firing rate increases from 120 kW/m2 to 240 KW/m2, the maximum gas temperature and the downstream gas temperature of the metal fiber increase by 6.21% and 8.55%, respectively, the outer surface temperature of the metal fiber increases by 6.42%, and the stable gas temperature downstream of the metal fiber reaches the maximum value at an equivalence ratio of 1. After the specimen is added, the internal and downstream gas temperatures of the metal fiber are significantly reduced, while the upstream gas and outer surface temperatures of the metal fiber show no significant changes. When the specimen diameter changes from 139mm to 98mm, the gas temperature and fiber surface temperature change is small, while the surface heat flux of the specimen significantly increases from 12.7 KW to 22.7 KW. It can be seen that when designing the test parameters, the surface heat flux of the specimen can be adjusted by adjusting the distance between the combustion surface and the specimen.

Formation Mechanism of Elemental Te Produced in Tellurite Glass Systems by Femtosecond Laser Irradiation.

The formation of elemental trigonal tellurium (t-Te) on tellurite glass surfaces exposed to femtosecondlaser pulses is discussed. Specifically, the underlying elemental crystallization phenomenon is investigated by altering laser parameters in common tellurite glass compositions under various ambient conditions. Elemental crystallization of t-Te by a single femtosecond laser pulse is unveiled by high-resolution imaging and analysis. The thermal diffusion model reveals the absence of lattice melting upon a single laser pulse, highlighting the complexity of the phase transformation. The typical cross-section displays three different crystal configurations over its depth, in which the overall thickness increases with each subsequent pulse. The effect of various controlled atmospheres shows the suppressing nature of the elemental crystallization, whereas the substrate temperature shows no significant impact on the nucleation of t-Te nanocrystals. This research gives new insight into the elemental crystallization of glass upon femtosecond laser irradiation and shows the potential to fabricate functional transparent electronic micro/nanodevices.

Heat transfer in rock masses: Application to the Lascaux Cave (France)

The conservation of prehistorical painted caves is a major issue for the history of mankind. Alterations, which can appear on walls, mainly result from the thermal convection taking place in caves. The convection in caves depends on the difference between the temperatures of the walls which mainly stem from the heat conduction in rock masses. In literature, this problem is currently solved with a uni-dimensional homogeneous model where the outside air temperature is the only source of energy. Here, based on the literature in other fields, we show that the ground surface temperature cannot be considered as equal to the outside temperature. Radiation, convection and water evaporation indeed figure prominently in the process. The mathematical guideline giving the topsoil temperature is detailed. In addition, we propose a mathematical assumption to ensure energy conservation. A simple experimental setup allowing the circumvention of this complex procedure is also suggested. Then, we apply the widely used uni-dimensional homogeneous thermal diffusion model, with the proposed ground surface temperature calculation, to the Lascaux Cave. The comparison between the measurements in the first meter below the ground surface and the calculated temperatures shows a good agreement. Within the cave, the model successfully predicts the temperature measured by a few thermocouples. For others, there are significant discrepancies, which seem to indicate that the homogeneous hypothesis is not precise enough to describe rock masses. The heterogeneity of the Lascaux Cave massif will thus need to be thermally investigated.

Experimental Study of Stretched Premixed Flame Stabilized in a Flat Channel near a Heated Wall

In this work, the behavior of a lean premixed stretched flame stabilized in a flat channel near a heated wall was studied. Dependences of the flame front position on the stretch rate parameter at temperatures of the heated wall of 1000 and 1200 K and the combustible mixture composition (ϕ = 0.7 and 0.6) were obtained experimentally. The reduced thermal diffusive model was used in numerical simulation for an explanation of obtained experimental results. Theoretical estimates are in qualitative agreement with the experiment. The performed qualitative analysis may be useful in estimation of the combustion product temperature and the residence time of the nanoparticles forming in combustion products before their impact with the hot wall.

Illumination and regolith temperature at China’s next candidate lunar landing site Shackleton crater

This study focuses on the illumination and temperature at China’s next lunar candidate landing site Shackleton crater. We used the NASA’s SPICE system to evaluate the terrain obscuration effect on real-time illumination; the resulting illumination map resembles previous studies, validating the methodologies used in our study. In addition, we estimated an accumulated illumination map for the period of likely rover movement. The map indicates the illuminated inner wall of the Shackleton crater is close to 27% of the whole, meaning that the rover will likely receive solar radiation during its movement. Using the real-time illumination and the distributed 1-D thermal diffusion model, we continuously evaluated the regolith temperature for more than 20 years to stabilize the temperature, and selected the temperature of the end time as the initial value used in a thermal study set for July 20, 2023 and May 8, 2027. Our results indicate the temperature in the permanent shadow region remains nearly constant, thus validating the stability of our estimated initial temperature. Our results also indicate that the surface temperature is more sensitive to transient illumination, but the subsurface temperature is more likely to be associated with the accumulated illumination. This difference indirectly implies that the conductivity of the lunar regolith is inefficient. The locations receiving more solar radiation show a temperature larger than the threshold (∼112 K) of ice stability. The permanently shadowed regions can be as cold as 25 K, and such extreme coldness is a hazard to the rover. There are suitable temperature locations which have a warm surface but cold subsurface to preserve water ice. To further ensure normal rover movement, we provided a map of suitable temperature sites and found that these locations exist not only in the Shackleton crater’s inner wall, but also outside the crater. We suggested four trade-off sampling sites with suitable temperatures and gradual slopes.

Alleviation of Self-Heating Effect in Top-Gated Ultrathin In2O3 FETs Using a Thermal Adhesion Layer

In this work, a thin layer of hexagonal boron nitride (h-BN) or HfO2 serves as an adhesion layer between the ultrathin atomic-layer-deposited (ALD) indium oxide (In2O3) channel and a sapphire substrate to enhance the thermal interfacial conductance. A thermo-reflectance (TR) measurement system with high spatial resolution is introduced to experimentally demonstrates the improvement. With the thin h-BN or HfO2 interlayer, the temperature elevation ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {T}$ </tex-math></inline-formula> ) induced by self-heating effect (SHE) is decreased by roughly 9% or 27%, respectively. To quantify the improvement of the interfacial heat transfer, a steady-state thermal diffusion model with a finite-element method is combined with the experimental TR observation to extract the effective thermal boundary conductance (TBC) values in each case. It is shown that the effective TBC is ameliorated by a factor of 2 or 7 with the h-BN or HfO2 interlayer, which is responsible for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {T}$ </tex-math></inline-formula> reduction. Furthermore, phonon density of states (PDOS) distribution mismatch implies that the intersection over union (IOU) ratio in the acoustic phonon region of In2O3 with h-BN or HfO2 is roughly 3 or 11 times higher than that directly with sapphire, which is responsible for the profound TBC enhancement. Based on this, ultrahigh maximum drain current of 2.4 mA/ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> is achieved with 2.1-nm-thick In2O3 channel on a sapphire substrate with an HfO2 interlayer in between due to the alleviated SHE.