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27-301: Microstructure-Properties: IL8: Thermal Properties

Fall 2007A. D. Rollett

Microstructure Properties

ProcessingPerformance

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• The objective of this lecture is to relate thethermal properties of materials to theirmicrostructure.

• A practical example is used of optimizationof thermal conductivity in electricallyinsulating materials (ceramics) formicroelectronic applications, e.g. SiC, Si3N4,diamond.

• Symbol for thermal conductivity: k or κ.

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Notation• k or κ is the thermal conductivity• <M> is the average atomic mass• Ω is the atomic volume• B is a constant• ΘD is the Debye temperature• T is the (absolute) temperature• γ is the Grüneisen constant• v is the speed of sound• l is the mean free path of phonons• C is the specific heat• ρ is the density.

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Thermal Conductivity• Thermal conductivity is driven by different

mechanisms in different types of materials.• Metals exhibit high electrical and thermal

conductivity as a consequence of easy transport ofelectrons.

• FCC metals exhibit the highest electrical andthermal conductivities, e.g. k(Ag)=430 W.m-1.K-1 atroom temperature. Alloying tends to decreaseconductivity. BCC metals typically exhibit an orderof magnitude lower electrical and thermalconductivity.

• Electrical conductivity is a very useful probe ofsolute levels and can be used to measure theprogress of precipitation, especially in Al alloys.

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Thermal Conductivity: Ceramics• Most ionically and covalently bonded materials exhibit very

low conductivities (electrical and thermal). This is a result ofthe large band gap between the valence and conductionbands.

• Covalently bonded solids tend to have higher thermalconductivities than ionically bonded solids.

• Insulating materials with high thermal conductivity are useful,however, for electronic applications where they are used toconduct heat away from components with high heatdissipation requirements.

• Adamantine materials (diamond-like) have thermalconductivities comparable to those of fcc metals.

• Thermal waves can be used to measure k, e.g. by irradiatinga sample with a laser pulse and observing the change intemperature on the other side.

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Microelectronic Packaging• All microelectronic circuits are placed on an insulating

substrate. In recent times, the power dissipation has becomea very important issue and high thermal conductivity is useful.

Basic package:D. Frear, JOM (1999), 51, 22-27 Proposed advanced technology:

scholar.lib.vt.edu/theses/available/ etd-0106100-113320/unrestricted/ChpI.pdf

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Heat Sinks forMicroelectronics

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Applications• Susceptors, rings for Rapid Thermal

Processing/Annealing• Plasma-etch components• RF-heated [radio frequency] susceptors• Optical components in synchrotrons and high lower

lasers• Electronics, especially LSI and VLSI (large scale

integration)• Wear resistant components

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Thermal Conductivity- basics• The kinetic theory of gases provides a basic model for which

thermal conductivity, k, depends on the specific heat, C, thevelocity, v, [of the carrier of heat] and the mean free path, l.

k = 1/3 C v l

• In solids, the heat carrier is the phonon and so the relevantmean free path is the distance between phonon scatteringcenters. Collisions between phonons are not relevantbecause they do not change the net momentum of thephonons involved in transport whereas so-called umklappprocesses, in which two phonons interact with the lattice toyield a third (scattered) phonon. These processes change thenet momentum of the phonons, precisely because of theinteraction with the lattice.

• Recall that sound velocities are related to elastic stiffness, E,and density, ρ:

v = √(E/ρ)

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Mean free path, specific heat• It turns out that at high temperatures, the number of

phonons involved in such scattering is proportionalto T, from which it follows that the mean free pathgoes as 1/T.

• Also, as the temperature decreases and the meanfree path goes up, eventually the length is limited byeither the specimen size or the grain size.

• At low enough temperatures,the specific heat varies stronglywith temperature, C ∝ T3. Thusthe thermal conductivity alsovaries at T3.

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High k Ceramics• Phonons are elastic waves in the crystal lattice.

Heat is mostly carried by phonons in the acousticrange, i.e. with frequencies below 20kHz (or 1000cm-1).

• Srivastava gives an equation for thermalconductivity at high temperatures from three-phonon(scattering) interactions, where <M> is the averageatomic mass, Ω is the atomic volume, B is aconstant, ΘD is the Debye temperature (~516K forAlN), T is the temperature, and γ is the Grüneisenconstant (~0.77 for AlN).

k = B<M>Ω1/3Θ3/(Tγ2)

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Example: AlN• Note the

differencebetween singlecrystalconductivity andthe (calculated)polycrystalbehavior.

• Conductivity firstincreases, thendecreases with T.

[Slack]

Ag

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High k Ceramics, contd.• Based on this equation, the following factors promote

high thermal conductivity.– Low atomic mass– Strong interatomic bonding (covalent in preference to ionic)

for high Debye temperature– Equivalently, high modulus– Simple crystal structure (less scattering of phonons)– Low anharmonicity– Isotopically pure

• Examples: Diamond 2000 (W.m-1.K-1) BN (cubic) 1300 SiC 490 BeO 370 AlN 320 BeS 300

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Material Dependence of k at RT• The independent

variable (x-axis) is<M>Ω1/3Θ3, asgiven in theprevious equation.

• Diamond has thehighest k.

[Slack]

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Defects• Heat sinks are typically made by sintering powders

together.• The important defects are the following.

– Point defects in the lattice– Grain boundaries– Pores– Second phases from dopants, sintering aids

• Grain size: just as in mechanical strength (Hall-Petch effect), the increase in thermal resistivity, k-1,is related to the the square root of the grain size,√d:

∆(k-1) ∝ √d

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Development of high k materials

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Point Defects• The importance of point

defects depends on thematerial under consideration.

• For example, in silicon nitride,the oxygen content is critical.

[Hirao et al., MRS Bulletin, June 2001, p451]

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Oxygen content• Oxygen content is

minimized bysintering under anitrogen atmosphere.

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Aligned Microstructures• Which direction has

the highestconductivity?Lowest?

• Si3N4 made bycombined seeding(with β-Si3N4) andtape casting.Sintering aids areY2O3 or Nd2O3,nitrogen atmosphere.

[Hirao et al., MRS Bulletin, June 2001, p451]

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Effect of Impurities in AlN• Oxygen contents

(in 1018 atoms.cm-3):

W201: 42.R47 (Si): 1R162: 300

• Conductivity goes downwith increasing oxygencontent

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CVDmicrostructures

• Conductivity scales with grainsize in the films made byChemical Vapor Deposition(CVD).

• Some anisotropy inconductivity (higherperpendicular to the filmplane) is apparent.Explicable in terms of thegrain structure, based onmean free path in differentdirections.

• Grain boundaries areeffective obstacles to heatflow.

[Goela et al., MRS Bulletin, June 2001, p458]

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CVD microstructures, contd.• Point defect density and dislocation density also

decrease with height in the film.

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Superconductivity• Another example of an (extreme) dependence of a transport

property is that of high temperature superconductivity.• The original HTc material, YBCO, is approximately tetragonal

and the superconducting property exists primarily in the a-bplane. Moreover, grain boundaries in this material aresignificant barriers to current flow (just as they are for heatflow).

Intl. Materials Reviews, 48 (2003), Kuppusami & Raghunathan

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Superconductivity, contd.• Consequently, in order to make long

lengths of superconductor (fortransmission cables, for example)enormous effort has gone intodevising ways of making highlyoriented thin films.

• The most successful methodinvolves producing a highly texturedmetal tape (e.g. Ni), depositingvarious buffer layers of oxides (e.g.CeO2) on the metal, and then finallythe superconductor.

“High-performance YBCO-coatedsuperconductor wires”, Paranthamanand Izumi, MRS Bulletin, 29 p533(2004)

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References• Physical Ceramics (1997), Y.-T. Chiang, D.P. Birnie III, W.D.

Kingery, Wiley, New York, 0-471-59873-9; pp. 474-477.• MRS Bulletin, June 2001, vol. 26, No. 6, “High Thermal

Conductivity Materials.”• “The Intrinsic Thermal Conductivity of AlN,” G.A. Slack, R.A.

Tanzilli, R.O. Pohl & J.W. Vandersande, J. Phys. Chem.Solids, 48, 641-647 (1987).

• JOM June 1998: special topic on “Thermal ManagementMaterials for Electronic Applications”, pp 46-72 (severalarticles).

• Electrons and Phonons, J.M. Ziman, Oxford, 1960.• ThermTile Ceramic Heat Sinks and Electrical Insulators From

CoorsTek, www.azom.com/details.asp?ArticleID=3729• http://www.customdicing.com/aln-links.htm• Heat Sink Materials and Components from WCu and MoCu.

AlN Ceramic: marketech-heatsinks.com/pages/aln.html

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Supplemental Slides

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Ceramic Microstructure• In order to understand the microstructures that we

are dealing with, it is necessary to delve intoceramic processing briefly (but see the course onCeramic Processing).

• Although sintering of powders can be performed onsingle-phase material, it is common to use sinteringaids that substantially modify the composition andmicrostructure of the final product.

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Sintering• Sintering is the name given to the process by which

powder compacts agglomerate when exposed tohigh temperature. The process is driven byreduction in surface area.

[Physical Ceramics]

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Geometry of Sintering

• Three stages of geometrical evolution:Initial/ neck formationIntermediate/ "channel" formation i.e.continuous, tubular porosityLate Stage/ closed, isolated porosity

• Estimate the relative densities at which the switch occurs from onestage to another based on monosize particles: channels exist oncethe edge length (all triple edges are pores) is >> pore diameter. Thissuggests a transition at 89% dense for simple cubic packing (if thepore channel diameter is less than the edge length divided by 3).The transition to late stage comes when pore channel diameter is sosmall that the channel breaks up into spherical pores. This is harderto estimate because kinetically determined.

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Sintering Mechanisms• We identify the five mass transport mechanisms,

w.r.t. initial stage (neck formation).1. Vapor transfer2. Boundary diffusion (grain boundary)3. Surface diffusion4. Bulk diffusion (notedifference insources of material)5. Dislocation diffusion

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Bond versus Shrink• We also need to identify the effect of each mechanism

because each one has a very different consequence ifdensification is desired, see fig. 5.40 in PhysicalCeramics. Clearly vapor phase transport is useful if aporous body is needed:

Vapor transfer Bonding, no shrinkageSurface diffusion Bonding, no shrinkageBulk diffusion Bonding if from surface

" " Shrinkage if from boundaryGrain Boundary diffusion ShrinkageDislocation diffusionShrinkage

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Intermediate Stage Sintering• The intermediate stage of sintering is the most complicated

and difficult to model rigorously. Morphologically, it is thestage between development of necks and the shrinking ofisolated pores. When the particles have necked to the pointthat they are faceted, fig. 5.42 in Physical Ceramics, thenthere are channels along the edge of each facet while theparticles are joined at the facets. At this stage, non-localbehavior can be very important. If plastic deformation,especially by creep, occurs at an appreciable rate, theparticles can deform and slide past each other.Rearrangement in space of particles can achieve a significantamount of densification without any requirement for diffusionor vapor transport. Most texts gloss over this stage ofsintering despite its importance.

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Late Stage Sintering• The driving pressure for late stage sintering is almost entirely

due to vacancy diffusion away from voids. For late stagesintering in particular, the temperatures typically used aresuch that grain boundary diffusion is dominant. Therefore thecritical feature of pore elimination is that the pores remainattached to grain boundaries. If the grain boundaries succeedin detaching from the pores, densification will essentiallycease. In other words, it is possible for grain coarsening totake place rapidly enough that boundaries become detachedfrom pores, leaving them behind within grains. Those isolatedpores do not lose volume as quickly as those on boundaries.

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Liquid Phase Sintering• Liquid phase sintering is effective because it accelerates mass

transport enormously. We need a small angle for liquid phasesintering in order that wetting occur efficiently and quickly. Itis critical that the liquid phase wet the solid phase, so thatwherever contacts have developed between particles, so thata grain boundary is present, the liquid can penetrate andreplace the g.b. with liquid. The criterion for this is simply that2γsl < γgb.

• Liquids also allow for particle rearrangement by effectivelylubricating the particles so that they can slide past oneanother.

• Elimination of voids is accelerated if the solid can dissolve inthe liquid phase, to a limited extent (obviously we do not wantcomplete dissolution). Diffusion rates in liquids are typicallythree to four orders of magnitude faster than in solids.

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Grain Growth vs. Densification• The elevated temperatures used for sintering are high enough

to allow grain growth to occur. This is generally undesirable inthe sintering process since pores attached to grain boundariescan shed vacancies and shrink whereas pores isolated in thegrain interiors depend on lattice (bulk) diffusion. At the thetemperatures typical of sintering, lattice diffusion is slowcompared to boundary diffusion. Therefore it is important thatgrain growth not occur so fast the the boundaries canseparate from the pores. Recall that grain growth is also athermally activated process described by the followingclassical description of the kinetics due to Hillert (1965), whereR is the grain size (diameter), M(T) is the grain boundarymobility as a function of temperature, t is time, and γ is thegrain boundary energy:

R2 - R02 = M(T)γt/8

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High Conductivity Ceramics• The previous slides summarized the situation in

standard ceramics processing in which high densitymust be combined with small grain size. In thecase of high conductivity ceramics, however, weneed large grain size in order to minimize both grainboundary scattering, and the effect of grainboundary phases.

• Therefore sintering aids to provide a liquid phaseduring sintering are commonly used. For siliconnitride, Yb2O3 and MgO are used, for example.

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Summary• Thermal conductivity has been reviewed• High thermal conductivity insulators (ceramics)

exist, such as diamond and AlN.• Thermal conductivity in insulators is dependent on

phonons: high conductivity requires minimization ofphonon scattering.

• Thermal conductivity is grain size dependent,provided that other defects (point defects,dislocations) do not limit it.

• Practical heat sinks (for microelectronics) requiresintering from powders. Therefore the properties ofthe sintered product is not as good as the singlecrystal properties.