8/2/2019 Com Sol Poster

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Comsol and Conductivity:

Modeling Heat TransferGeneral Heat Transfer

General PDE Form

Ballistic-Diffusive Equations

Looking Forward

Sarah Palaich and Prof. Brian Daly Vassar College Summer 2007

 As the size of microprocessors steadily decreases into the nanoscale,

the study of heat transfer in nanostructures is critical. We decided toapproach this phenomenon not only from within the lab, but also usinga computer simulation. We used a graphical finite modeling program,Comsol, to further investigate heat flow. Comsol uses a CAD 3Dgraphics modeler and built in partial differential equations to simulatevarious types of physics. We began by modeling a sample with a series of 20µm diameter

aluminum dots on silicon substrate that was already in the lab. Thephoto to the left is an image of that wafer. The aluminum dots wereplaced at varying distances from each other, but for the Comsol model

we used a center to center distance of 20µm and reduced the diameter

to 14µm. We started modeling heat flow at 300K, but quickly realized thatthermal conductivity, the key property that determines heat flow, is toolow at 300K to create measurable detection using the ultrafast lasersetup. We switched to modeling at 100K and also decided to simulatea bridge of silicon rather than the silicon on insulator composition foundin the samples we have in the lab. Once the temperature and structural changes were made we wereable to see measurable heat flow. In addition, we began to modify theequations given by Comsol to better model the actual behavior of heatflow in nanosctrues.

  The Comsol program comes with build in modules. One of these isheat transfer, which is set up to deal w ith both conductive andconvective heat dispersion.  The basis for almost all models of heat transfer outside of thenanoscale realm is the basic partial differential diffusion equation. Thisequation depends on the material properties density, specific heat andthermal conductivity. The equation is exclusively “diffusive” in natureand has no ballistic, wave-like, properties.  The image shows the Comsol solution for heat flow at 100K using theheat transfer equation built into the module. The left aluminum dot isinitially heated to 110K and the heat flows into silicon and over to theother dot in 50 nanoseconds. The temperature in the second dotincreases by about 0.2K. Although the heat transfer module allows for the modification of material properties, the equations are set. To create a better model of 

heat flow in nanostructures, we needed to move beyond the basicdiffusion equation.

 In order to have the optimum control over our model, werecreated the diffusion equation using Comsol's general partialdifferential equation (PDE) form. Most of the terms of the general PDE form are unnecessary. Infact, in its simplest form we only used c and d

a. We set c = 1 to

keep the second derivative of position and da

is merely the

inverse of diffusivity, a constant related to density, heatcapacity and thermal conductivity. We used this new equation form on the silicon bridge only. TheAl dots and the SiO

2and Si layers continued to use the general

heat transfer equation.  The most important step of this process was making the PDEtalk to the general heat transfer equations at the domainboundaries. We managed this using thermal boundaryresistance conditions that had the two different temperaturevariables relate to each other. Even though the new PDE only modeled heat flow as a diffusiveprocess, we now had the ability to add complexity.

 Heat flow in nanostuctures deviates substantially from thesame process in bulk materials. This is due to boundaryscattering encountered by the heat carrying phonons. As thestructures decrease in size, the phonons are more likely toencounter boundaries and scatter than in larger bulk materials.In terms of material properties, this decreases the thermalconductivity and is the root of cooling problems in modernmicroprocessors. Numerous theoretical approximations have been made tomodel this new behavior. The most popular is the Boltzmanntransport equation, but that equation is difficult to solve forstructures of complex geometry. A 2002 paper by Gang Chen explains the concept of ballistic-diffusive equations that more simply model the behavior of phonons in nanostructures. At right is Chen's breakdown of heat flow into ballistic anddiffusive components. The artificial wave front in the diffusivecomponent is especially interesting since it is an effect of approximating the behavior with the diffusion equation. In fact,the diffusion equation assumes that heat transfer can beinstantaneous, which is untrue. We used Chen's ideas as a starting point for our approximationsof heat flow as both ballistic and diffusive.

 By the end of our summer, we understood the value of combining both the ballistic anddiffusive properties of heat flow into a Comsol model. As we continue our study, we want tocreate a model that integrates these elements together such that we can eventually test thebehavior of similar samples in the laser lab. As a start we are now looking at the telegraph equation, shown to the left. The telegraphequation has sometimes been called the one dimensional heat flow equation when thesecond derivative of time is removed (top). As illustrated by the second equation, thesecond derivative of time, a hallmark of wave behavior, appears. The equation is easilytranslated into Comsol's general PDE form. We have already created one and two dimensional models utilizing the telegraph equation.At present we are confronted with difficulties at the boundaries, the most critical surfaces.Nevertheless, we hope to be able to soon have a model that has ballistic and diffusiveproperties complementing each other. An ultimate goal is to have a functional 3D bridge model like the ones we constructed for

both general heat flow and PDE diffusive heat flow. This requires multiple PDE domainstalking to each other through complex boundary conditions.In the lab, we would like to have our cryostat fully functional so w e can start testing theaccuracy of Comsol's predictions at 100K for the diffusive heat flow models. We would alsolike to be able to work with Al dots connected by Si bridges such as those in the models.

Invaluable SourcesPhonons 101/100/111

 Phonons are the equivalent of photons for sound; they are quantizedvibrations in crystals. Of course, for heat transfer we are dealing with veryhigh frequency phonons, so we cannot hear anything.  The energy carried by phonons is recognized by temperature increase. Within nanostructures the scattering of phonons at the boundaries of thesubstance becomes especially important and causes heat to transfer in adifferent manner than in bulk samples.  The general behavior of phonon movement, and therefore heat transport isdiffusive. This begins to change as the mean free path of the phononsbecomes comparable with the size of the structure within which they aretraveling In essance, the phonon environment has gone from a large gymwith a few basketball teams playing to a tiny hallway with most of the team

members still present. As we continue to study smaller structures, actual phonon behaviorbecomes even more important to understand. Indeed, the little quantizedvibrations lead the way into the mysterious nanoscale realm. Weunderstand the atomic and macroscopic, but the nanoscale is the nextunexplored frontier.

C ∂T ∂ t 

∇⋅−∇ T =Q k ∂2

u∂  x 2

=∂u∂ t k = C

Comsol heat transfer equationρ = densityC = specific heatκ = thermal conductivity T = temperature

Diffusion equationρ = densityC = specific heatκ = thermal conductivityu = temperature

ea ∂

2u∂ t 2da ∂

u∂ t  ∇⋅−c∇ u− u B⋅∇ uau=f 

General Comsol partial differential equationc = diffusion coefficienta = absorption coefficientf = source terme

a= mass coefficient

da

= damping/mass coefficient

α = conservative flux convection coefficientβ = convection coefficientγ = conservative flux source term

∂

2um

∂t 2∂ um

∂ t =∇

C∇ um−∇⋅qbq̇e

∂ q̇e

∂ t 

Diffusion with ballistic element qb

∂2u

∂ x 2=1

∂u∂ t 

  Telegraph equationκ = thermal conductivityu =temperature

∂2u

∂ t 2∇⋅−c

2∇ uuut =0

Comsol telegraph equationα,β = constantsC = propagation velocity

Asheghi, M., Leung, Y., Wong, S., and Goodson, K., 1997, “Phonon-boundaryScattering in Thin Silicon Layers,” Appl. Phys. Lett., 71, pp. 1798-1800.

Cahill, D., Goodson, K. and Majumdar, A., 2002, “Thermometry and Thermal Transport in Micro/Nanoscale Solid-State Devices and Structures,” Journal of Heat Transfer, 124, pp. 223-241.

Chen, G., 2002, “Ballistic-Diffusive Equations for Transient Heat Conductionfrom Nano to Macroscales,” Journal of Heat Transfer, 124, pp. 320-328.

Sneddon, I.N., 1957, Elements of Partial Differential Equations , McGraw-Hill,New York.

 The Si bridge using a PDE to model the diffusion equation with the Al dots using general heat transfer.

Si bridge and Al dots using Comsol's general heat transfer equation.

Chen's comparison of diffusive and ballistic components of heat flow.

Image of the Al dots on SOI (silicon on insulator) wafer.

Modeling Heat Flow in Nanosctrures