Chapter 11: Microscale Conduction
11.1 Introduction
•
Microscale heat conduction is dramatically different than at the macroscale
•
This becomes obvious when bulk values for thermal conductivity are compared with
thermal conductivity of microsamples
•
Reference values for thermal conductivity given in handbooks are bulk values
•
An example comparing bulk and microscale values for the thermal conductivity of
silicon at various temperatures is given
•
The majority of microscale systems follow the pattern illustrated in the silicon example:
microscale samples have smaller values of thermal conductivity than their bulk
counterparts
•
There are systems that exhibit the opposite trend (i.e. carbon)
•
The reduced thermal conductivity at the microscale can be both an obstacle (i.e. heat
dissipation in microelectronics) and a desirable property (i.e. thermal barrier coatings)
11.1.1 Categories of Microscale Phenomena
•
Classical Fourier law of diffusive heat conduction breaks down for processes that are too
fast and for systems that are too small
•
Chapter Focus: Small systems at steady state
•
Essential Questions
•
What are the physical mechanisms by which the classical macroscopic Fourier’s law
will fail for small systems?
•
At what length scale does this happen?
•
How can we modify the macroscopic Fourier’s law to still be useful at the
microscale? What is the effective thermal conductivity k for a microstructure such as
a nanowire or thin film?
•
“Wavepacket” concept is used to provide framework (See Fig. 11.1.a)
•
A “wavepacket” is a wave-like disturbance that is localized within a small volume of
space and has both wave-like and particle-like properties
•
•
Particle-like due to small size and propagation direction
•
Wave-like in that it has an average wavelength λ
When the characteristic length L of a material is much larger than the mean free path Λ
between wavepacket collisions, L >> Λ >> λ , the wavepacket can be treated like a
particle
•
When L ≥ Λ >> λ , the wavepacket is still approximated as a particle, but undergoes
numerous collisions with the system boundaries that further impede energy flow – this is
the classical size effect
•
When L ≈ λ , the wave nature of the wavepacket cannot be ignored – this is the quantum
size effect, which requires a more sophisticated analysis
•
Note: Fig. 11.1 illustrates the various size effects discussed
•
The classical size effect is the focus of this chapter
11.1.2 Purpose and Scope of this Chapter
•
Chapter purpose: introduce key concepts of the classical size effect in steady-state heat
conduction at the microscale
•
Supporting subjects such as solid state physics, quantum mechanics and statistical
thermodynamics are not addressed
11.2 Understanding the Essential Physics of Thermal Conductivity Using Kinetic
Theory of Gases
•
The kinetic theory approach offers maximum physical insight for minimal complexity
•
It can be applied to a wide range of realistic problems in microscale conduction
•
The results of more complicated analyses, such as Boltzmann transport equation and
Landauer formalism, can still be understood using kinetic theory
11.2.1 Derivation of Fourier’s Law and an Expression for the Thermal Conductivity
•
Individual molecules in an ideal gas exchange energy and change direction during
collisions
•
Collisions happen repeatedly over time, causing each particle to undergo a “random
walk” between collisions
•
The distance traveled between collisions is the free path; the average of the free paths
over time and over all particles is the “mean free path” and is denoted by Λ
•
The corresponding mean free time is denoted by τ and defined as:
Λ = vτ
(11.1)
where v is the speed of the particles
•
See Figure 11.2 (a-c) for corresponding illustrations
•
Derivation of Fourier’s Law by net energy flow evaluation
•
Consider Fig. 11.2 (d), in which a gas of particles is placed in a large box in which
the temperature decreases from left to right (in the x direction)
•
•
Now consider a plane x = x 0 in the middle of the box, in the middle of two control
volumes each with thickness Λ in the x-direction and with some large area A in the yz directions
•
Left volume extends from x0 to x0 – Λ
•
Right volume extends from x0 to x0 + Λ
After a time τ = Λ / v approximately half of the particles have exited through the
boundaries of each control volume
•
The particles from the left are hotter than the particles from the right
•
Note: In the following derivation, it is more convenient to work with energy per unit
volume [J/m3] than energy per unit mass [J/kg]
•
The total energy in the left and right control volumes, respectively, after time τ are
Uˆ left = ΛAU left , avg
Uˆ right = ΛAU right , avg
•
As a result of energy leaving each control volume, the net energy crossing x = x 0 is
ΔUˆ LR ≈ 12 Uˆ left − 12 Uˆ right = 12 ΛA(U left , avg − U right , avg )
•
Assuming U is a smoothly-varying function of x that can be approximated as a
straight line over a distance on the order of Λ
U left , avg ≈ U (x 0 − 12 Λ )
U right , avg ≈ U (x 0 + 12 Λ )
•
Using a Taylor series expansion:
⎡ dU ⎤
U left , avg − U right , avg ≈ − ⎢
⎥ Λ
⎣ dx ⎦ x0
(11.2)
•
This assumes local thermodynamic equilibrium, in which the energy density of the
particles conforms to the local temperature
•
Exact analysis shows the correct expression is 2/3 as large as eq. (11.2)
⎡ dU ⎤
U left , avg − U right , avg = − 23 ⎢
⎥ Λ
⎣ dx ⎦ x0
•
•
The rest of the chapter uses the correct 2/3 coefficient
Using the chain rule, results can be presented in terms of temperature using specific
heat capacity at constant volume: C = ρ cv
where C is in units J/m3-°C
•
Collecting results and dividing by the area and elapsed time to get heat flux in W/m2
yields:
q ′′ =
•
•
ΔUˆ LR
dT
= − 13 CvΛ
Aτ
dx
(11.3)
Thus, Fourier’s law of heat conduction was derived
Additionally, an equation for the thermal conductivity of a gas of particles was derived:
k = 13 CvΛ
•
To understand heat conduction in any system, each term in eq. (11.4) needs to be
methodically evaluated
•
In microscale systems, the small size of the microstructure will generally reduce Λ
without affecting C or v, leading to reductions of thermal conductivity
(11.4)
11.3 Energy Carriers
•
Relevant properties C, v and Λ will be discussed for a range of heat-conducting materials
•
Key results from kinetic theory are directly applicable to solids, provided the
corresponding “particles” that carry heat can be identified
•
This allows analysis to be extended from ideal gases to metals, insulators and
semiconductors, and to heat transfer by radiation
•
Same three questions are asked for each identified energy carrier:
•
What is C?
•
What is v?
•
What is Λ?
11.3.1 Ideal Gases: Heat is Conducted by Gas Molecules
•
Recall the ideal gas law
pV = mtot RT
(11.5)
where
p is absolute pressure
V is volume
mtot is the total mass of the gas
T is temperature in absolute units [K]
R is the gas constant for the particular gas in question, found from
R = RU M
where
(11.6)
RU = 8.314 J/mol-K is the universal gas constant
M is the molecular weight of the gas molecule
RU is also conveniently expressed as:
RU = N A k B
(11.7)
where
N A = 6.022 × 10 23 mol −1 is Avagadro’s number
k B = 1.381 × 10 −23 J/K is Boltzmann’s constant
•
Note: The need to use absolute units is a recurring theme in this chapter and use of
absolute units is strongly encouraged throughout
•
Results for specific heat, speed and mean free path for monatomic gases are discussed,
since monatomic gases can only store kinetic energy in translational kinetic energy
•
Connections of results to diatomic/polyatomic gasses are briefly commented on
•
•
Diatomic and polyatomic gases can store kinetic energy in rotational motion as well
as interatomic vibrations within gas molecules, making analysis more complicated
Specific Heat
•
For ideal gas, specific heats are related through
c p = cv + R
•
[ J/kg - °C]
This is converted to volume by multiplying by the density: ρ = p / RT
C p = Cv + p / T
•
[J/m 3 - °C]
(11.8)
For monatomic ideal gas, specific heat has the form:
C v = 32 R
Cv =
•
•
3p
2T
(11.9)
Specific heats of diatomic and polyatomic gases have similar form, but the numerical
coefficient is a function of temperature that is larger or equal to 3/2
Speed
•
Molecular speeds are distributed over a broad range of values, even at fixed pressure
and temperature
•
Distribution of speeds is temperature dependent and is known as the Maxwellian
velocity distribution
•
For simplicity, this is represented by a single “thermal velocity” vth represented by
the root-mean-square velocity v rms
vth = vrms = 3RT
•
(11.10)
Most-probable speed ( v mp = 2 RT ), the mean speed ( v mean =
8
π
RT ), and the speed
of sound ( v sound = γRT , where γ = c p / c v ) are sometimes used instead of the rootmean-square velocity
•
•
RT
These are all proportional to
Mean Free Path
•
Mean free path between collisions for gas molecules is
Λ coll =
1
π 2η d 2
where
η = ρN A / M = p / k BT is the average number of molecules per unit volume
d is the effective diameter of the gas molecue
•
When molecules collide, they no dot exchange 100% of their energy, so the mean
free path for energy exchange is given to within 2% by
Λ en =
•
⎛ 12
= ⎜⎜
5 3π η d
⎝ 5 3π
12
2
⎞ k BT
⎟⎟
2
⎠ pd
(11.11)
•
This assumes the molecules of ideal monatomic gases are elastic spheres
•
Note that Λ en is about 3.5 times larger than Λ coll , and that Λ en is the correct choice
for evaluating thermal conductivity
An example (example 11.1) is presented in which the thermal conductivity for Helium
gas at 0°C and atmospheric pressure is calculated and compared with experimental
results
11.3.2 Metals: Heat is Conducted by Electrons
•
“Free electrons” are responsible for high electrical conductivity in metals
•
They also carry energy along with charge and thus are the dominant transporter of heat
in metals
•
Speed
•
Electrons in a metal that contribute to heat transfer travel at the same speed, the
Fermi velocity, v F
•
Typical values are around 1 − 2 × 10 6 m/s
•
Values for various metals are found in solid state physics handbooks and
textbooks
•
Free electrons are also characterized by their Fermi energy E F : E F = 12 me v F2 where
me = 9.110 × 10 −31 kg is the mass of an electron
•
Fermi velocity is also related to the concentration of free electrons η e
vF =
(
h
3π 2η e
me
)
1/ 3
(11.12)
where h = 1.055 × 10 −34 J - s
•
Specific heat
•
Specific heat of free electrons, C e , follows a simple equation that can be expressed
in three equivalent ways:
C e = 12 π 2η e k B2T / E F
C e = 12 π 2η e k B T / TF
(11.13)
C e = γT
•
The second form defines a characteristic “Fermi Temperature”
TF = E F / k B
•
The third form expresses the fact that the specific heat is simply proportional to
Temperature
•
•
•
•
(11.14)
The coefficient γ has units of J/m3-K2
Note that the electron specific heat is typically several orders of magnitude smaller
than the standard handbook values for C for metals, even though the electrons
dominate the thermal conductivity
Mean free path
•
Electron scattering in metals is complex
•
Primary mechanism for electron scattering at room temperature and above is
collisions with sound waves
•
In practice, electron mean free path is inversely proportional to temperature at 300 K
and above
•
Typical electron mean free path in metals is of the order of tens of nm at room
temperature
Table 11.1 tabulates free electron properties of several metals, including the Fermi
velocity and specific heat coefficient
11.3.3 Electrical Insulators and Semiconductors: Heat is Conducted by Phonons
(Sound Waves)
•
Insulators (dielectrics) have extreme scarcity of free electrons as compared to metals
•
Thermal energy is stored and transported in dielectric crystals by atomic vibrations
•
The simplest and most important class of atomic vibrations is sound waves
•
Sound waves have wavelengths that are much larger than the spacing between atoms in
the crystalline structure
•
Sound wavelength λ and frequency ω follow the relationship:
ω=
2π v s
λ
(11.15)
where v s is the speed of sound in the material
•
Note the similarity to the equation for the frequency of a light wave
•
A “phonon” is the quantum of a sound wave in the same way that a “photon” is the
quantum of a light wave.
•
There are two types of phonons: acoustic and optical
•
Acoustic Phonons
•
•
Largely thought of as sound waves following equation (11.15)
•
There is an upper limit on the allowed frequencies
•
Phonon wavelengths become shorter with higher frequencies
•
At sufficiently high frequency, wavelengths become comparable to the lattice
constant
•
It is unphysical to speak of wavelengths shorter than twice the interatomic
spacing in the lattice, hence the upper limit on allowed frequencies
•
Maximum frequencies are about 1013 – 1014 rad/s
•
If details of phonon behavior at high frequency are important, “lattice dynamics”
is used
Optical Phonons
•
Optical phonons are present if and only if the material’s crystal structure has more
than one atom per “primitive unit cell”
•
Face-centered cubic (FCC), body-centered cubic (BCC), and simple cubic crystals
all have primitive unit cells consisting of one atom
•
Dynamics of optical phonon vibrations are dominated by oscillations of atoms
locally against their nearest neighbors
•
Local oscillations between atoms of different charges lead to a local magnetic field
that oscillates over time, making the crystals capable of interacting strongly with
light at certain frequencies
•
Oscillation frequencies are slightly higher than the maximum frequencies of acoustic
phonons
•
The oscillation frequencies are relatively insensitive to changes in phonon
wavelength
•
This leads to the fact that optical phonons propagate at velocities far slower than the
speed of sound – so slowly that the velocity of optical phonons is commonly set to
zero
•
The approximation of zero velocity implies that optical phonons make a negligible
contribution to thermal conductivity
•
Table 11.3 compares acoustical and optical phonons
•
Optical phonons are ignored for the rest of this chapter while discussing thermal
conductivity
•
Chapter focus: Acoustic phonons and the Debye approximation
•
Speed
•
Acoustic phonons are approximated as traveling at the speed of sound in the material
•
Crystals have different sound speeds for transverse and longitudinal waves and there
are twice as many transverse waves as longitudinal waves
•
The transverse and longitudinal wave speeds, v s ,T and v s , L , respectively, are
combined into an effective sound velocity when discussing thermal conductivity in
nanostructures:
vs =
•
•
(
2
3
v s−,T2 + 13 v s−,2L
)
−1 / 2
(11.16)
Averaging in this inverse-squares sense ensures exact results at low temperature
Specific Heat
•
The Debye model gives an integral expression for specific heat that requires
numerical methods for evaluation
•
A practical and convenient approximation that is exact in the limits of low and high
temperature is
C=
3η PUC k B
5 ⎛θD ⎞
1+
⎜ ⎟
4π 4 ⎝ T ⎠
(11.17)
3
where
η PUC is the number of primitive unit cells per unit volume
θ D is the “Debye temperature” given by
θD =
•
(
hv s
6π 2η PUC
kB
)
1/ 3
(11.18)
This can be used with less than 2% error compared to the exact Debye equation
•
•
A slightly different definition of θ D is in common use; it is identical to eq.
(11.18) except that η PUC is replaced with η atoms , which is the number density of
atoms
•
This is equivalent for crystals with one atom per unit cell
•
For atoms with more than one atom per unit cell, this results in larger
temperatures
•
This is inappropriate for modeling specific heat, since it treats optical
phonons as if they were traveling at the speed of sound
Equation (11.17) reduces to the following expressions in the low and high
temperature limits:
⎛T
12π 4
C=
η PUC k B ⎜⎜
5
⎝θD
C = 3η PUC k B
•
•
⎞
⎟⎟
⎠
3
(T < 203 θ D )
(11.19)
(T > 12 θ D )
(11.20)
•
Eq. (11.19) is exact for T / θ D → 0 and can be used up to T ≈
than 20% compared to the exact Debye equation
•
Eq. (11.20) is exact for T / θ D → ∞ and can be used down to T ≈ 12 θ D with errors
less than 20%
•
Eq. (11.19) is known as the “Debye T 3 law”
•
The high temperature result eq. (11.20) is known as the “Law of Dulong and Petit”
•
Optical phonons make their own contribution to specific heat, which is described
using an “Einstein model”
•
Values of specific heat reported in handbooks are for total specific heat, which
includes optical and acoustic phonons
3
20
θ D with errors less
Mean Free Path
•
At room temperature and above, phonon scattering is dominated by phonons
scattering with other phonons that the overall energy flux is impeded; this is
“Umklapp” scattering
•
In practice, phonon scattering is approximately inversely proportional to temperature
at room temperature and above
•
Alloy atoms can also result in strong scattering
•
Dopant atoms can scatter phonons
•
At temperatures around 300 K and below it may also be important to consider effects
of phonons scattering off of impurities, isotopes, defects and grain boundaries
Example (11.2) is presented, in which a power law approximation for thermal
conductivity for silicon between 300 K and 1000 K is produced
•
Example (11.3) is presented, in which the mean free path for silicon as a function of
temperature is estimated
•
Figure (11.3) in this example summarizes several key ideas and trends, including
high and low temperature trends for specific heat, thermal conductivity and mean
free path
11.3.4 Radiation: Heat is Carried by Photons (Light Waves)
•
Using kinetic theory in the microscale perspective, radiation and conduction can be
viewed as two limiting cases of a single general phenomenon
•
The equivalence becomes clearer in the sections on boundary scattering and classical
size effect
•
Radiation is treated as a gas of photons
•
Speed
•
•
The speed of light in a vacuum is a physical constant: c = 2.998 × 10 8 m/s
Specific heat
•
Photons store energy, just as molecules, electrons and phonons do
•
In a vacuum chamber at absolute temperature T, the higher the temperature, the
greater the number of photons inside the chamber and the greater the energy per
photon on average
•
In a perfect vacuum inside the chamber, specific heat is given by
C = 16σ T 3 / c
(11.21)
where σ = 5.670 × 10 −8 W/m 2 - K 4 is the Stefan-Boltzmann constant
•
•
•
This is regardless of the emissivity of the chamber walls
Mean free path
•
Mean free path varies tremendously, and is usually much larger than that of
molecules, electrons and phonons
•
Photon mean free path usually depends strongly on wavelength and material, with
typical values of Λ in the range of microns to millimeters
Example (11.4) is presented, in which the effective thermal conductivity for radiation
heat transfer between two parallel plates is calculated
•
The following equations are derived within the example:
•
Net radiation heat transfer from gray plate “1” to black plate “2”:
(
Q = εσA T14 − T24
where
ε is the emissivity of plate 1
)
(11.22)
A is the area of each plate
T1 is the temperature of plate 1
T2 is the temperature of plate 2
•
“Conduction resistance”
R=
T1 − T2
1
=
Q
4εσAT 3
where T is the average temperature: T =
•
1
2
(11.23)
(T1 + T2 )
Effective mean free path for photons within the problem conditions:
Λ eff =
12εσ T 3 L
= 34 εL
16σ T 3 / c v
(
)
(11.24)
where L is the distance between the plates
•
Note that in eq. (11.24) c = v
11.4 Thermal Conductivity Reduction by Boundary Scattering: The Classical Size
Effect
•
The previous sections have focused on bulk materials, so thermal conductivity is
independent of exact size and shape of the sample, so
k bulk = 13 CvΛ bulk
•
For traditional applications with sample sizes ranging from mm to m, the bulk mean free
path is several orders of magnitude smaller than the sample, so handbook values of
thermal conductivity are appropriate
•
In modern technologies, where the characteristic length of a structure is easily 10-100
nm, energy carriers collide frequently with the boundaries of the structure
•
This is the classical size effect, which is explored in this section
11.4.1 Accounting for Multiple Scattering Mechanisms: Matthiessen’s rule
•
Thermal conductivity is still expressed as
k = 13 CvΛ eff
•
The speed and specific heat of the energy carriers in the nanostructures are approximated
as identical to the values in a bulk sample
C eff ≈ C bulk
•
and v eff ≈ vbulk
Efforts are fully focused on determining Λ eff
•
Matthiessen’s rule is a powerful approximation for calculating the effective mean free
path that can be adapted to represent the results of more detailed solutions found from
the Boltzmann transport equation
•
This approximation has good accuracy and provides excellent physical insight and is
widely used
•
Matthiessen’s rule states that the effective mean free path is the sum of the mean free
paths corresponding to each of the various scattering mechanisms, summed in a
reciprocal sense:
Λ−eff1 =
•
∑Λ
−1
i
mechanism i
(11.25)
The sum includes as many terms as there are scattering mechanisms
•
Example for phonon scattering on impurities, grain boundaries, electrons, other
phonons and sample boundaries:
1
1
Λ−eff1 = Λ−imp
+ Λ−1.b. + Λ−ph1 −e + Λ−ph1 − ph + Λ−imp
1444g 4
4244444
3
1
Λ−bulk
•
For simplicity, scattering mechanisms are partitioned into two categories
•
Mechanisms present in a large, bulk sample Λ bulk
•
Mechanisms due to “boundary scattering” Λ bdy
•
Matthiessen’s rule is thus expressed as
(
1
1
Λ eff = Λ−bulk
+ Λ−bdy
)
−1
=
Λ bulk Λ bdy
Λ bulk + Λ bdy
(11.26)
11.4.2 Boundary Scattering for Heat Flow Parallel to Boundaries
•
Figure 11.4 (a) shows standard configurations for heat flow parallel to boundaries,
including nanowires and thin films
•
Nanowires
•
For a nanowire with diameter D, the mean free path due to boundary scattering is
given by
⎛1+ p ⎞
⎟⎟
Λ bdy = D⎜⎜
⎝1− p ⎠
where p is the specularity and is defined as
p = 0 : scattering is 100% diffuse (rough surfaces)
p = 1 : scattering is 100% specular (smooth surfaces)
0 < p < 1 : scattering is p specular and (1 − p ) diffuse
(11.27)
•
Specular scattering refers to mirror-like reflections where the angle of incidence
equals the angle of reflection
•
Diffuse scattering refers to reflections off a rough surface such that all incident
particles are reflected with equal intensity in all directions regardless of incident
angle
•
See Figure 11.5 for an illustration of specular and diffuse scattering
•
For surface roughness smaller than wavelength, reflections will be speculare
•
For roughness larger than wavelength, reflections will be diffuse
•
For intermediate roughness, specularity is estimated using the following expression
⎛ − 16π 3δ 2 ⎞
⎟⎟
p = exp⎜⎜
λ2
⎝
⎠
(11.28)
where
δ is the root-mean-square roughness
λ is the wavelength of the energy carrier
•
•
At low temperatures, wavelengths of most carriers become significantly longer than
their 300 K values
Thin Films (Diffuse: p = 0)
•
Analysis of heat transport along a thin film with thickness L is best done using
Boltzmann transport equation, but results can be expressed in kinetic theory
framework using effective mean free paths
•
For purely diffuse scattering, the standard result is the Fuchs-Sondheimer solution:
Λ eff
Λ bulk
= 1−
⎛ L
3Λ bulk ⎡
⎢1 − 4 E 3 ⎜⎜
8L ⎣
⎝ Λ bulk
⎞
⎛ L
⎟⎟ + 4 E5 ⎜⎜
⎠
⎝ Λ bulk
⎞⎤
⎟⎟⎥
⎠⎦
(11.29)
where E3 and E5 are “exponential integrals,” special functions defined through
E n ( x ) = ∫ μ n − 2 exp(− x / μ )dμ
1
0
where μ is a dummy variable of integration
•
Figure 11.6 shows the exponential integral functions
•
The following asymptotic forms of (11.29) are useful for thick and thin films,
respectively
Λ eff
Λ bulk
⎛ 3Λ bulk ⎞
= ⎜1 +
⎟
8L ⎠
⎝
−1
L >> Λ bulk
(11.30a)
Λ eff
Λ bulk
•
3L ln (Λ bulk / L )
L << Λ bulk
4Λ bulk
(11.30b)
•
Eq. (11.30a) has errors less than 10% for L > 0.5Λ bulk and errors less than 20% for L
as small as 0.025Λ bulk
•
Eq. (11.30b) has errors less than 10% for L < 0.019Λ bulk and errors less than 20%
for L as large as 0.125Λ bulk
Thin Films (Some specularity: p > 0)
•
Effective thermal conductivity increases when film specularity becomes significant
•
Effective mean free path is given by
Λ eff
Λ bulk
•
=
= 1−
3Λ bulk (1 − p ) 1
1 − exp(− L / μΛ bulk )
μ − μ3
dμ
∫
0
2L
1 − p exp(− L / μΛ bulk )
(
)
(11.31)
•
This is readily evaluated numerically, and resulting mean free paths for a range of
specularities are shown in Fig. 11.7
•
Note that the p = 0 case correctly reduces to (11.29), while for p = 1 there is no
reduction in k compared to bulk value, regardless of L
Example 11.5 is presented, in which the specularity and effective thermal conductivity
for a silicon nanowire at 300 K are calculated
•
The following equation for thermal conductivity results
k
k bulk
•
=
1
1 + (Λ bulk / Λ bdy )
=
1
1 + (Λ bulk / D )
(11.32)
This results from Λ bdy = D , which results from the specularity being approximately
zero
•
•
The problem uses Λ bulk = 78 nm ; a better estimate is Λ bulk ≈ 200 − 300 nm , which
accounts for the frequency dependence of C and v
Example 11.6 is presented, in which thermal conductivity as a function of temperature is
plotted for nanowires of several diameters
•
The plots in the example illustrate how the thermal conductivity varies as sample
size decreases, showing that at 20 K thermal conductivity is 6000 times smaller for a
56 nm diameter nanowire than for a bulk sample
•
At higher temperatures, a more careful analysis using the Born-von Karman
approach gives better agreement between calculated thermal conductivity and
experimental results
11.4.3 Boundary Scattering for Heat Flow Perpendicular to Boundaries
•
Figure 11.4 (b) shows standard configurations for heat flow perpendicular to boundaries
•
Kinetic theory is in general less appropriate for these configurations as compared to
transport parallel to boundaries, but still gives useful physical insight that is accurate in
simple cases
•
Thin films with no heat generation
•
For heat transport perpendicular to a thin film of thickness L without heat generation,
heat flow is described using effective thermal conductivity and Matthiessen’s rule as
long as the mean free path is expressed as
Λ bdy =
3
4
L
α + α 2−1 − 1
−1
1
(11.33)
where α 1 and α 2 represent the absorptivities of the two bounding surfaces
•
Note that the concept of emissivity and absorptivity from photon radiation has been
generalized to other energy carriers
•
In the context of heat conduction by gases, absorptivities are replaced by “energu
accommodation coefficients
•
The assumption that emissivity and absorptivity are approximately equal is made
(Kirchoff’s Law)
•
Eq. (11.33) is one expression of the Rosseland diffusion approximation with Deissler
jump boundary conditions; this is commonly derived in the field of radiation heat
transfer in a participating medium or using the Boltzmann transport equation
•
Example 11.7 is presented, in which the effective thermal conductivity as a function
of film thickness L is plotted, as well as conductive thermal resistance as a function
of L for an area A of 1 cm by 1 cm
•
The following equations are derived:
•
Thermal conductivity as a function of L
k (L ) = 13 CvΛ eff =
•
1
4Λ bulk
1+
3L
(11.35)
Thermal resistance as a function of L with constant A:
R (L ) =
•
(11.34)
Rewitten as:
k (L )
=
k bulk
•
CvΛ bulk
CvΛ bulk L
=
4Λ bulk 3L + 4Λ bulk
1+
3L
1
3
L L + 43 Λ bulk
=
kA
k bulk A
(11.36)
The sequence of plots in Fig. 11.9 is an excellent example of the classical
size effect
•
The thick-film limit, L >> Λ bulk , is known as the “diffusive” regime because
the thermal resistance is dominated by the diffusion of energy carriers from
the hot side to the cold side
•
•
In this case, thermal resistance is linearly proportional to film thickness
The thin-film limit, L << Λ bulk , is known as the “ballistic” regime
•
k is proportional to L
•
R becomes independent of L
•
There are no scattering effects within the film itself and thermal
resistance is dominated by emissivity and absorptivity, as it is in
traditional blackbody radiation resistance for photons
•
Transition between diffusive and ballistic regimes occurs in the example
when the film thickness is approximately equal to the mean free path
•
From Matthiessen’s rule, more generally, transition from diffusive to ballistic
regimes occurs when the bulk mean free path is comparable to the boundary
scattering mean free path
•
See Table 11.5 for a comparison between diffusive and ballistic limits
11.5 Closing Thoughts
Summary
•
•
•
Kinetic theory framework was used to introduce the essential concepts of conduction
heat transfer by various energy carriers
•
For any choice of energy carrier and nanostructure, the key tasks are the same:
•
Determine the specific heat C, the carrier velocity v and the mean free path Λ eff
The chapter is limited to the classical size effect
•
Wavepacket is approximate as a particle
•
Characteristic length L may be larger than, comparable to, or much smaller than the
bulk mean free path Λ bulk
An important recurring theme is the transition from diffusive to ballistic behavior
Suggestions for further study
•
Improving kinetic theory by integrating over frequency
•
Frequency dependence can be accounted for by writing equation (11.4) as an integral
over frequency:
k (T ) =
•
1
3
∫ Cω (ω , T )v(ω )Λ (ω , T )dω
eff
(11.36)
Arguments in parenthesis indicate the functional dependencies of the various quantities
•
Alternative theoretical techniques
•
The Boltzmann transport equation is the origin of several solutions previously presented
•
Perspective of radiation heat transfer in a participating medium is also useful
•
Explicit link between these approaches is the “equation of radiative transfer”
•
Superlattices
•
Solutions for heat transfer in superlattices are generally derived from Boltzmann
transport equation
•
Ultrafast phenomena
•
Important if the characteristic time of a process is shorter than or comparable to the
mean free time
•
The mean free time is also known as the relaxation time
•
The order of magnitude of the mean free time depends on the energy carrier and the
temperature