13.3.1 Spreading Resistance In chip packages that provide for lateral spreading of the heat generated in the chip, the increasing cross-sectional area for heat flow in the “layers” adjace
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suggested that when the grouping,NT/φ, where N is the numberof atomic layers in
the thin film,T the temperature, and φ the Debye temperature, is less than unity, the
heat capacity can be expected to display sensitivity to the characteristic dimension
Their results show that such size effects on the thermodynamic properties are more important at cryogenic temperatures
13.3.1 Spreading Resistance
In chip packages that provide for lateral spreading of the heat generated in the chip, the increasing cross-sectional area for heat flow in the “layers” adjacent to the chip reduces the heat flux in successive layers and hence the internal thermal resistance
Unfortunately, however, there is an additional resistance associated with this lateral flow of heat, which must be taken into account in determination of the overall chip package temperature difference The temperature difference across each layer of such
a structure can be expressed as
where
or
For the circular and square geometries common in microelectronic applications, Negus et al (1989) provide an engineering approximation for the spreading resistance
R spof a small heat source on a thick substrate or heat spreader, insulated on the sides and held at a fixed temperature along the base as
R sp= (0.475 − 0.626 + 0.13ζ)3
whereζ is the square root of the heat source area divided by the substrate area, k the
thermal conductivity of the substrate, anda the area of the heat source.
The spreading resistance R sp from eq (13.22) can now be added to the one-dimensional conduction resistance to yield the overall thermal resistance of that layer
It is to be noted that the use of eq (13.22) requires that the substrate be three to five times thicker than the square root of the heat source area Consequently, for relatively thin layers on thicker substrates, such as thin lead frames or heat spreaders interposed between the chip and the substrate, eq (13.22) cannot be expected to provide an acceptable prediction ofR sp Instead, use can be made of the numerical results plotted
in Fig 13.6 to obtain the appropriate value of the spreading resistance
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Figure 13.6 Spreading resistance on thin layers
Kennedy (1959) analyzed heat spreading from a circular uniform heat flux source
to a cylindrical substrate with isothermal temperature boundaries at the edges, the bottom, or both, and presented the results in graphic form Spreading resistance charts prepared originally by Kennedy (1959) were reproduced by Sergent and Krum (1994) Although the boundary conditions and geometry assumed by Kennedy (1959)
do not match the mixed boundary conditions and rectangular shapes found in most electronic packages, the spreading resistance results can be used with acceptable accuracy in many design situations (Simons et al., 1997) Using the spreading re-sistance factorH from the appropriate Kennedy graph, the spreading resistance is
calculated using
R sp = H
wherek is the thermal conductivity and a is the heat source area (Figs 13.7 to 13.9).
Song et al (1994) developed an analytical model to estimate the constriction–
spreading thermal resistanceR spfrom a circular or rectangular heat source to a sim-ilarly shaped convectively cooled substrate Lee et al (1995) extended the solutions provided by Song et al (1994) to present closed-form expressions for dimensionless constriction–spreading thermal resistance based on average and maximum tempera-ture rise through the substrate Their set of equations is
R sp,avg= 0.5(1 − ζ)3/2φc
k√Asource
(13.24)
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R sp,max=
1/√π(1 − ζ)3/2φc
k√Asource
(13.25)
φc= tanhλc τ + (λ c · Bi)
λc= π + 1
τ = δ
Bi=hr
Figure 13.7 Spreading resistance factor H1forsurfacez = w at zero temperature (From
Simons et al., 1997.)
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0.01 0.1 1.0 10
a b/
H1
w
b a
z
T = 0
0.01 0.015 0.02 0.025 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00 2.50 3.00 4.00 5.00 6.00 7.00 8.00 9.00
10.00w b/
r
Figure 13.8 Spreading resistance factor H2forsurfacez = w at zero temperature (From
Simons et al., 1997.)
In eqs (13.24)–(13.29),R sp,ave is the constriction resistance based on the average
source temperature,R sp,maxthe constriction resistance based on the maximum source
temperature,δ the fin thickness, r the outer radius of the substrate, h the convective
heat transfer coefficient, andk the thermal conductivity The authors claim the
ex-pressions to be accurate to within 10% for a range of source and substrate shapes and for source and substrate rectangularity aspect ratios less than 2.5 Use of the convective boundary condition on the substrate base makes these relations especially
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Figure 13.9 Spreading resistance factor H3forsurfacez = w at zero temperature (From
Simons et al., 1997.)
well suited to the analytical determination of the spreading resistance of a chip-to-heat sink assembly as well as a chip encapsulated in a convectively cooled plastic package
13.3.2 Heat Flow across Solid Interfaces
Heat transfer across an interface formed by the joining of two solids is accompanied
by a temperature difference caused by imperfect contact between the two solids
Even when perfect adhesion is achieved between the solids, the transfer of heat is impeded by the acoustic mismatch in the properties of the phonons on either side of the interface Traditionally, the thermal resistance arising due to imperfect contact
has been called the thermal contact resistance The resistance due to the mismatch
in the acoustic properties is usually termed the thermal boundary resistance The
thermal contact resistance is a macroscopic phenomenon, whereas thermal boundary resistance is a microscopic phenomenon
Thermal Contact Resistance When two surfaces are joined, as shown in Fig
13.10, asperities on each of the surfaces limit the actual contact between the two
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Figure 13.10 Contact and heat flow at a solid–solid interface
solids to a very small fraction, perhaps just 1 to 2% for lightly loaded interfaces, of the apparent area As a consequence, the flow of heat across such an interface involves solid-to-solid conduction in the area of actual contact,Aco, and conduction through the fluid occupying the noncontact area,A nc, of the interface At elevated tempera-tures or in vacuum, radiation heat transfer across the open spaces may also play an important role The pressure imposed across the interface, along with microhardness
of the softer surface and the surface roughness characteristics of both solids, deter-mine the interfacial gapδ and the contact area A co Assuming plastic deformation of the asperities and a Gaussian distribution of the asperities over the apparent area, for the contact resistanceR co, Cooperet al (1969) proposed
R co= 1.45 k s (P/H )0.985
−1
(13.30)
wherek sis the harmonic mean thermal conductivity, defined ask s = 2k1k2/(k1+k2);
P the apparent contact pressure; H the hardness of the softer material; and σ the
root-mean-square (rms) roughness, given by
whereσ1andσ2are the roughness of surface 1 and 2, respectively The term|tan θ|
in eq (13.30) is the average asperity angle:
|tan θ|2= | tan θ1|2+ | tan θ2|2 (13.32) This relation neglects the heat transfer contribution of any trapped fluid in the inter-facial gap
In the pursuit of a more rigorous determination of the contact resistance, Yovano-vich and Antonetti (1988) found it possible to predict the area-weighted interfacial gap,Y , in the form
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Y = 1.185σ
− ln3.132P
H
0.547
(13.33)
whereσ is the effective rms as given by eq (13.31), P the contact pressure (Pa), and
H the surface microhardness (Pa) of the softer material, to a depth of the order of
the penetration of the harder material UsingY as the characteristic gap dimension
and incorporating the solid–solid and fluid gap parallel heat flow paths, Yovanovich (1990) derived for the total interfacial thermal resistance,
R co=
1.25k s|tan θ|
σ
P
H
0.95
+k Y g
−1
(13.34)
where k g is the interstitial fluid thermal conductivity In the absence of detailed information,σ/|tan θ| can be expected to range from 5 to 9 µm forrelatively smooth
surfaces
Thermal Boundary Resistance When dealing with heat removal from a chip and thermal transport in various packaging structures at room temperature and above, the thermal boundary resistanceR bis generally negligible compared to the contact re-sistance However, at the transistor level, where interfaces—often formed by epitaxial thin film deposition, through atomistic processes such as physical vapor deposition—
may be nearly perfect, the thermal boundary resistance should be included Two theoretical models are widely used to predict the thermal boundary resistanceR b:
the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM) The
former is based on the specular reflection of sound waves at the interface and the latter
is based on the diffuse scattering of phonons at the interface Swartz and Pohl (1989) provide a comprehensive discussion of both AMM and DMM models for the thermal boundary resistance and have shown that the microscopic thermal boundary resis-tance resulting from the mismatch in the acoustic properties in the low-temperature limit can be obtained by the following DMM equation:
R b= π2
30
k4
b
¯
h3
jj c−2
1,jj×j c−2
2,jj
jj c−2
1,jj+jj c−2
2,jj
wherek bis the Boltzmann constant, ¯h the Planck constant divided by 2π, c the speed
of sound,jj the mode of sound (jj = 1 forthe longitudinal mode, and jj = 2 forthe
transverse mode), and the subscripts 1 and 2 refer to the two solids in contact Note that this relation is strictly valid only at very low temperatures Similar equations
to estimate R b using DMM at high temperatures or AMM can be obtained from Swartz and Pohl (1989) Although AMM and DMM are based on very different physical arguments, they appear to yield identical results for most material pairs (Swartz and Pohl, 1989) Both these models are very good in predictingR b at very
low temperatures but fail miserably at high temperatures for various reasons, such
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as increased scattering of phonons and deviation from the Debye density of states (Swartz and Pohl, 1989)
Interstitial Materials In describing heat flow across an interface, eq (13.34) assumed the existence of a fluid gap, which provides a parallel heat flow path to that of the solid–solid contact Because the noncontact area may occupy in excess
of 90% of the projected area, heat flow through the interstitial spaces can be of great importance Consequently, the use of high-thermal-conductivity interstitial materials, such as soft metallic foils and fiber disks, conductive epoxies, thermal greases, and polymeric phase-change materials, can substantially reduce the contact resistance
The enhanced thermal capability of many of high-performance epoxies, thermal greases, and phase-change materials commonly in use in the electronic industry is achieved through the use of large concentrations of thermally conductive particles
Successful design and development of thermal packaging strategies thus requires the determination of the effective thermal conductivity of such particle-laden interstitial materials and their effect on the overall interfacial thermal resistance
Comprehensive reviews of the general role of interstitial materials in controlling contact resistance have been published by several authors, including Sauer (1992)
When interstitial materials are used for control of the contact resistance, it is desirable
to have some means of comparing theireffectiveness Fletcher(1972) proposed two parameters for this purpose The first of these parameters is simply the ratio of the logarithms of the conductances, which is the inverse of the contact resistance, with and without the filler:
χ = lnκcm
in whichκ is the contact conductance, and cm and bj refer to control material and bare
junctions respectively The second parameter takes the thickness of the filler material into account and is defined as
η = (κδfiller) cm
in whichδ is the equivalent thickness
The performance of an interstitial interface material as decided by the parameter defined by Fletcher(1972), in eqs (13.36) and (13.37) includes the bulk as well as the contact resistance contribution It is for this reason that in certain cases the thermal resistance of these thermal interface materials is higher than that for a bare metallic contact because the bulk resistance is the dominant factor in the thermal resistance (Madhusudan, 1995) To make a clear comparison of only the contact resistance arising from the interface of the substrate and various thermal interface materials,
it is important to measure it exclusively Separation of the contact resistance and bulk resistance will also help researchers to model the contact resistance and the bulk resistance separately
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Equations (13.36) and (13.37) by Fletcher (1972), show that the thermal resistance
of any interface material depends on both the bond line thickness and thermal con-ductivity of the material As a consequence, for materials with relatively low bulk conductivity, the resistance of the added interstitial layer may dominate the thermal behavior of the interface and may result in an overall interfacial thermal resistance that
is higherthan that of the bare solid–solid contact (Madhusudan, 1995) Thus, both the conductivity and the achievable thickness of the interstitial layer must be considered
in the selection of an interfacial material Indeed, while the popular phase-change materials have a lower bulk thermal conductivity (at a typical value of 0.7 W/m · K)
than that of the silicone-based greases (with a typical value of 3.1 W/m · K), due to
thinner phase-change interstitial layers, the thermal resistance of these two categories
of interface materials is comparable
To aid in understanding the thermal behavior of such interface materials, it is useful
to separate the contribution of the bulk conductivity from the interfacial resistance, which occurs where the interstitial material contacts one of the mating solids Fol-lowing Prasher (2001), who studied the contact resistance of phase-change materials (PCMs) and silicone-based thermal greases, the thermal resistance associated with the addition of an interfacial material,RTIMcan be expressed as
RTIM= Rbulk+ R co1+ R co2 (13.38) whereRbulkis the bulk resistance of the thermal interface material, andR cothe contact
resistance with the substrate, and subscripts 1 and 2 refer to substrates 1 and 2 Prasher (2001) rewrote eq (13.38) as
RTIM= δ
κTIM
2κTIM
Anom
Areal
2kTIM
Anom
Areal
(13.39)
whereRTIM is the total thermal resistance of the thermal interface material,δ the
bond-line thickness,κTIMthe thermal conductivity of the interface material,σ1 and
σ2the roughness of surfaces 1 and 2, respectively,Anomthe nominal area, andAreal the real area of contact of the interface material with the two surfaces Equation (13.39) assumes that the thermal conductivity of the substrate is much higher than that of the thermal interface material The first term on the right-hand side of eq
(13.39) is the bulk resistance, and other terms are the contact resistances Figure 13.11 shows the temperature variation at the interface between two solids in the presence of
a thermal interface material associated with eq (13.39) Unlike the situation with the more conventional interface materials, the actual contact area between a polymeric material and a solid is determined by capillary forces rather than surface hardness, and an alternative approach is required to determineArealin eq (13.39) Modeling each of the relevant surfaces as a series of notches, and including the effects of surface roughness, the slope of the asperities, the contact angle of the polymer with each the substrates, the surface energy of the polymer, and the externally applied pressure, a surface chemistry model was found to match very well with the experimental data for PCM and greases at low pressures (Prasher, 2001), as shown in Fig 13.12 for
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Figure 13.11 Temperature drops across an interface
PCM Unfortunately, it has not yet been found possible to determine the contact area with a closed-form expression It is also to be noted that eq (13.39) underpredicts the interface thermal resistance data at high pressures
Thermal Conductivity of Particle-Laden Systems Equation (13.39) shows that the bulk and contact resistance of the thermal interface material are dependent
on the thermal conductivity of the interface material The thermal conductivity of
a particle-laden polymer increases nonlinearly with increasing volume fraction of the conducting particle, as suggested in Fig 13.13 One of the most commonly used models for predicting the thermal conductivity of a particle-laden, two-phase system is the Lewis and Nielsen (1970) model This model calculates the thermal conductivity of two-phase system using