18.2.5 Heat Capacity The rate of thermal transport within a material is governedby the thermal diffusivity, which is the ratio of the thermal conductivity to the heat capacity.. An under
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公4K
M
1
Figure 18.4 Plot of the frequency of a plane wave propagating in the crystal as a function of
wavevector Note that the relationship is linear until k 1/a.
whereM is the mass of an individual atom By taking the time dependence of the
solution to be of the form exp(−iωt), the frequency of the solution as a function of
the wavevector can be determined as given by eq (18.13) Figure 18.4 shows the results of this equation plotted over all the values that produce independent results
Values of k larger thanπ/a correspondto plane waves with wavelengths less than
the interatomic spacing Because the atoms are locatedat discrete points, solutions
to the equations above yielding wavelengths less than the interatomic spacing are not unique solutions, andthese solutions can be equally well representedby long-wavelength solutions
ω(k) =
4K
M
sin12ka
The results shown in Fig 18.4 apply for a Bravais lattice in one dimension, which can be representedby a linear chain of identical atoms connectedby springs with the same spring constant,K A Bravais lattice with a two-point basis can be represented
in one dimension by either a linear chain of alternating massesM1andM2, separated
by a constant spring constantK, or by a linear chain of constant masses M, with the
spring constant of every other spring alternating betweenK1andK2 The theoretical results are similar in both cases, but only the case of a linear chain with atoms connectedby two different springs,K1andK2, where the springs alternate between the atoms, is discussed The results are shown in Fig 18.5 The displacement of each atom from each equilibrium point is given by eitheru(na) for atoms with the K1 spring on their right andv(na) for atoms with the K1spring on their left
The reason for selecting this case is its similarity to the diamond structure Recall that the diamond structure is a FCC Bravais lattice with a two-point basis; all the atoms are identical, but the spacing between atoms varies As the distance varies
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K1 K2
a
u na ( ) v na( )
( )a
( )b
R= na
Figure 18.5 (a) One-dimensional Bravais lattice with two atoms per primitive cell shown in
their equilibrium positions The atoms are identical in mass; however, the atoms are connected
by springs with alternating strengthsK1andK2 (b) One-dimensional Bravais lattice with two
atoms per primitive cell where the atoms are displaced byu(na) and v(na).
between atoms, so do the intermolecular forces, which are represented here by two different spring constants The equations of motion for this system are given by
M d dt2u2n = −K1 (u n − v n ) − K2(u n − v n−1 ) (18.14a)
M d dt2v2n = −K1 (v n − u n ) − K2(v n − u n+1 ) (18.14b)
whereu nandv nrepresent the displacement of the first andsecondatoms within the primitive cell, andK1andK2are the spring constants of the alternating springs Again taking the time dependence of the solution to be of the forme −ıax, the frequency of
the solutions as a function of the wavevector can be determined as given by eq (18.15) andshown in Fig 18.6, assuming thatK2> K1:
ω2= K1+ K2
1
M
K2
1 + K2
2+ 2K1 K2cos ka (18.15) The expression relating the lattice vibrational frequencyω andwavevector k is
typically calledthe dispersion relation There are several significant differences
be-tween the dispersion relation for a Bravais lattice without a basis [eq (18.13)] versus
a lattice with a basis [eq (18.15)] One of the most valuable pieces of information that can be gatheredfrom the dispersion relation is the group velocity The group velocityv ggoverns the rate of energy transport within a material andis given by the expression
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Optical
Acoustic
公
2K
M2
2(K K)
M
2K
M1
( )k
0
Figure 18.6 Dispersion relation for a one-dimensional Bravais lattice with a two-point basis
The dispersion relations shown in Fig 18.4 and in the lower curve in Fig 18.6 are
both roughly linear until k 1/a, at which point the slope decreases and vanishes
at the edge of the Brillouin zone, where k = π/a From these dispersion relations
it can be observedthat the group velocity stays constant for small values of k and
goes to zero at the edge of the Brillouin zone It follows directly that waves with
small values of k, corresponding to longer wavelengths, contribute significantly to the
transport of energy within the material These curves represent the acoustic branch of
the dispersion relation because plane waves with small k, or long wavelength, obey a
linear dispersion relationω = ck, where c is the speedof soundor acoustic velocity.
The upper curve shown in Fig 18.6 is commonly referredto as the optical branch
of the dispersion relation The name comes from the fact that the higher frequencies associatedwith these vibrational modes enable some interesting interactions with light at or near the visible spectrum The group velocity of these waves is typically much less than for the acoustic branch Therefore, contributions from the optical branch are usually considered negligible when evaluating the transport properties
Contributions from the optical branch must be considered when evaluating the spe-cific heat
Dispersion relations for a three-dimensional crystal in a particular direction will look very similar to one-dimensional relations except that there are transverse modes
The transverse modes arise due to the shear waves that can propagate in a three-dimensional crystal The two transverse modes travel at velocities slower than the longitudinal mode; however, they still contribute to the transport properties The optical branch can also have transverse modes Again, optical branches occur only in three-dimensional Bravais lattices with a basis and do not contribute to the transport properties, due to their low group velocities
Figure 18.7 shows the dispersion relations for lead at 100 K (Brockhouse et al., 1962) This is an example of a monoatomic Bravais lattice, since leadhas a face-centeredcubic (FCC) crystalline structure Therefore, there are only acoustic branches, one longitudinal branch and two transverse In the [110] direction it is
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Figure 18.7 Dispersion relation for leadat 100 K plottedin the [110] and[100] directions
(From Brockhouse et al., 1962.)
possible to distinguish between the two transverse modes; however, due to the sym-metry of the crystal, the two transverse modes happen to be identical in the [100]
direction (Ashcroft and Mermin, 1976)
Finally, the concept of phonons must be introduced The term phonon is commonly
used in the study of the transport properties of the crystalline lattice The definition
of a phonon comes directly from the equation for the total internal energyU l of a
vibrating crystal:
U l =
k,s
n s (k) +1
2 ¯hω(k, s) (18.17)
The simple explanation of eq (18.17) is that the crystal can be seen as a collection
of 3N simple harmonic oscillators, where N is the total number of atoms within the system and there are three modes of oscillation, one longitudinal and two trans-verse Using quantum mechanics, one can derive the allowable energy levels for a simple harmonic oscillator, which is exactly the expression within the summation
of eq (18.17) The summation is taken over the allowable phonon wavevectors k
andthe three modes of oscillations The definition of a phonon comes from the
fol-lowing statement: The integer quantityn s (k) is the mean number of phonons with
energy ¯hω(k, s) Therefore, the number of phonons at a particular frequency simply
represents the amplitude to which that vibrational mode is excited Phonons obey the Bose–Einstein statistical distribution; therefore, the number of phonons with a particular frequencyω at an equilibrium temperature T is given by the equation
n s (k) = e¯hω(k,s)/k1
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wherek B is the Boltzmann constant Most thermal engineers are familiar with the concept of photons Photons also obey the Bose–Einstein distribution; therefore, there are many conceptual similarities between phonons andphotons
The ability to calculate the energy storedwithin the lattice is important in any analysis of microscale heat transfer Often, the calculations, which can be quite cum-bersome, can be simplifiedby integrating over the allowable energy states These integrations are actually performedover frequency, which is linearly relatedto en-ergy through Planck’s constant The specific internal enen-ergy of the lattice,u l, is then given by the equation
u l =
s
D s (ω)n s (ω)¯hω s ∂ω s (18.19)
whereD s (ω) is the phonon density of states, which is the number of phonon states
with frequency betweenω and (ω + dω) for each phonon branch designated by s.
The actual density of states of a phonon system can be calculated from the measured dispersion relation; although often, simplifying assumptions are made for the density
of states that produce reasonable results
18.2.5 Heat Capacity
The rate of thermal transport within a material is governedby the thermal diffusivity, which is the ratio of the thermal conductivity to the heat capacity The heat capacity
of a material is thus of critical importance to thermal performance In this section the heat capacity of crystalline materials is examined An understanding of the heat ca-pacity of the different energy carriers, electrons and phonons, is important in the fol-lowing section, where thermal conductivity is discussed The heat capacity is defined
as the change in internal energy of a material resulting from a change in temperature
The energy within a crystalline material, which is a function of temperature, is stored
in the free electrons of a metal andwithin the lattice in the form of vibrational energy
Electron Heat Capacity To solve for the electron heat capacity of a free electron metal,C e, the derivative of the internal energy, stored within the electron system, is taken with respect to temperature:
C e= ∂u ∂T e = ∂T ∂
∞ 0
εD(ε)f (ε) dε (18.20)
The only temperature-dependent term within this integral is the Fermi–Dirac distribu-tion Therefore, the integral can be simplified, yielding an expression for the electron heat capacity:
C e= π2k B2n e
whereC eis a linear function of temperature andn eis the electron number density.
The approximations made in the simplification of foregoing integral hold for electron temperatures above the melting point of the metal
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Phonon Heat Capacity Deriving an expression for the heat capacity of a crystal
is slightly more complicated Again, the derivative of the internal energy, stored within the vibrating lattice, is taken with respect to temperature:
C l= ∂u l
∂T =
∂
∂T
s
D s (ω)n s (ω)¯hω s ∂ω
(18.22)
To calculate the lattice heat capacity, an expression for the phonon density of states
is required There are two common models for the density of states of the phonon
system, the Debye model and the Einstein model The Debye model assumes that all
the phonons of a particular mode, longitudinal or transverse, have a linear dispersion relation Because longer-wavelength phonons actually obey a linear dispersion rela-tion, the Debye model predominantly captures the effects of the longer-wavelength
phonons In the Einstein model, all the phonons are assumedto have the same
fre-quency andhence the dispersion relation is flat; this assumption is thus more repre-sentative of optical phonons Because both optical andacoustic phonons contribute
to the heat capacity, both models play a role in explaining the heat capacity However, the acoustic phonons alone contribute to the transport properties; therefore, the Debye model will typically be used for calculating the transport properties
relation is linear andall three acoustic branches have the speedof soundc:
However, unlike photons, this dispersion relation does not extend to infinity Since there are only N primitive cells within the lattice, there are only N independent
wavevectors for each acoustic mode Using spherical coordinates again, conceive
of a sphere of radiusk D in wavevector space, where the total number of allowable wavevectors within the sphere must beN and each individual wavevector occupies a
volume of(2π/L)3:
4
3πk3
D = N
2π
L
3
→ k D=
6π2N V
1/3
(18.24)
Using eq (18.24), the maximum frequency allowedby the Debye model, known as
the Debye cutoff frequencyωD, is
ωD = c
6π2N V
1/3
(18.25)
Now that the maximum frequency allowedby the Debye model is known andit is assumed that the dispersion relation is linear, an expression for the phonon density
of states is required Again, the concept of a sphere in wavevector space can be usedto findthe number of allowable phonon modesN with wavevector less than k.
Each allowable wavevector occupies a volume in reciprical space equal to(2π/L)3
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Therefore, the total volume of the sphere of radiusk must be equal to the number of
phonon modes with wavevector less thank multipliedtimes (2π/L)3:
4
3πk3= N
2π
L
3
→ N = V
6π2k3 (18.26) The phonon density of statesD(ω) is the number of allowable states at a particular
frequency andcan be determinedby the expression
D(ω) = ∂N ∂ω= V
2π2c3ω2 (18.27) Returning to eq (18.22), all the information needed to calculate the lattice heat capacity is known Again simplifying the problem by assuming that all three acoustic modes obey the same dispersion relation,ω(k) = ck, the lattice heat capacity can be
calculatedusing
C l (T ) = 3V ¯h2
2π2c3k B T2
ωD 0
ω4 e¯hω/k B T
(e¯hω/k B T − 1)2dω (18.28)
which can be simplifiedfurther by introducing a term calledthe Debye temperature,
θD The Debye temperature is calculateddirectly from the Debye cutoff frequency,
k BθD = ¯hω D → θD = ¯hω D
With this new quantity, the lattice specific heat calculatedunder the assumptions of the Debye model can be expressed as
C l (T ) = 9Nk B
T
θD
3 θD/T
0
x4e x
(e x − 1)2 dx (18.30) Figure 18.8 shows the molar values of the specific heat of Au comparedto the re-sults of eq (18.30) using a Debye temperature of 170 K Although a theoretical value
of the Debye temperature can be calculatedusing eq (18.29), the publishedvalues are typically determined by comparing the theoretical predictions of the specific heat
to measuredvalues
The low-temperature specific heat is important in the analysis of the lattice thermal conductivity If the temperature is much less than the Debye temperature, the lattice heat capacity is proportional toT3 This proportionality is easily seen in Fig 18.9, where the information containedin Fig 18.8 is plottedon a logarithmic plot to highlight the exponential dependence on temperature The Debye model accurately predicts thisT3dependence
the assumption that the dispersion relation is flat In other words, the assumption is
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Figure 18.8 Molar specific heat of Au comparedto the Debye model (eq 18.30) using 170
K for the Debye temperature (From Weast et al., 1985.)
made that allN simple harmonic oscillators are vibrating at the same frequency, ω0; therefore, the density of states can be written as
The methodfor calculating the heat capacity is exactly the same as that followed with the Debye model, although the integrals are simpler, due to the delta function
0.0001
0.01 0.1
0.001
1
100 10
Temperature (K)
K)
Figure 18.9 TheT3dependence of the lattice specific heat is very apparent on a logarithmic plot of the molar specific heat of Au (Weast et al., 1985), comparedto the Debye model (eq
18.30) using 170 K for the Debye temperature
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This model provides better results than the Debye model for elements with the di-amondstructure One reason for the improvement is the optical phonons in these materials Optical phonons have a roughly flat dispersion relation, which is better representedby the Einstein model
18.2.6 Thermal Conductivity
The specific energy carriers have been discussed in previous sections The manner
in which these carriers store energy, andthe appropriate statistics that describe the energy levels that they occupy, have been presented In the next section we focus
on how these carriers transport energy andthe mechanisms that inhibit the flow of thermal energy Using very simple arguments from the kinetic theory of gases, an expression for the thermal conductivityK can be obtained:
K = 1
whereC is the heat capacity of the particle, v the average velocity of the particles,
andl the mean free path or average distance between collisions.
Electron Thermal Conductivity in Metals Thermal conduction within metals occurs due to the motion of free electrons within the metal According to eq (18.32), there are three factors that govern thermal conduction: the heat capacity of the energy carrier, the average velocity, andthe mean free path As shown in eq (18.21), the electron heat capacity is linearly relatedto temperature As for the velocity of the electrons, the Fermi–Dirac distribution, eq (18.7), dictates that the only electrons within a metal that are able to undergo transitions, and thereby transport energy, are those locatedat energy levels near the Fermi energy The energy containedwith the electron system is purely kinetic andcan therefore be convertedinto velocity Because all electrons involvedin transport of energy have an amount of kinetic energy close to the Fermi energy, they are all traveling at velocities near the Fermi velocity Therefore, the assumption is made that all the electrons within the metal are traveling at the Fermi velocity, which is given by
v F =
2
The thirdimportant contributor to the thermal conductivity is the electron mean free path, obviously a direct function of the electron collisional frequency Electron collisions can occur with other electrons, the lattice, defects, grain boundaries, and
surfaces Assuming that each scattering mechanism is independent, Matthiessen’s
rule states that the total collisional rate is simply the sum of the individual scattering
mechanisms (Ziman, 1960):
νtot = νee+ νep+ νd+ νb (18.34)
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where νee is the electron–electron collisional frequency, νep the electron–lattice collisional frequency,νd the electron–defect collisional frequency, andνb the elec-tron–boundary collisional frequency Consideration of each of these scattering mech-anisms is important in the area of microscale heat transfer
The temperature dependence of the collisional frequency can also be very im-portant Electron–defect and electron–boundary scattering are both typically inde-pendent of temperature, whereas for temperatures above the Debye temperature, the electron–lattice collisional frequency is proportional to the lattice temperature Elec-tron–electron scattering is proportional to the square of the electron temperature:
νee AT2
whereA and B are constant coefficients and T e andT l are the electron andlattice
temperatures In clean samples at low temperatures, electron–lattice scattering dom-inates However, electron–lattice scattering occurs much less frequently than simple kinetic theory wouldpredict In very pure samples andat very low temperatures, the mean free path of an electron can be as long as several centimeters, which is more than 108times the distance between lattice sites This is because the electrons do not scatter directly off the ions, due to the wavelike nature that allows the electrons to travel freely within the periodic structure of the lattice Scattering occurs only when there are disturbances in the periodic structure of the lattice
The temperature dependence of the thermal conductivity often allows us to iso-late effects from several different mechanisms that affect the thermal conductivity
Figure 18.10 shows the thermal conductivity of three metals commonly used in the microelectronics industry: Cu, Al, and W The general temperature dependence of all three metals is very similar At very low temperatures, below 10 K, the primary
Figure 18.10 Thermal conductivity of Cu, Al, andW plottedas a function of temperature
(From Powell et al., 1966.)
... obtained:K = 1
whereC is the heat capacity of the particle, v the average velocity of the particles,
andl the mean free path or average distance... calculate the lattice heat capacity is known Again simplifying the problem by assuming that all three acoustic modes obey the same dispersion relation,ω(k) = ck, the lattice heat capacity can... determined by comparing the theoretical predictions of the specific heat
to measuredvalues
The low-temperature specific heat is important in the analysis of the lattice thermal conductivity