Microscale conduction is dramatically different from macroscale conduction Example: Thermal...

• Microscale conduction is dramatically different from macroscale conduction

• Example: Thermal conductivity in silicon

Bulk Silicon•Characteristic lengths range from meters to microns•At room temperature

k =148 W/m-°C•At 20 K

k = 4940 W/m-°C

Silicon Nanowire•Diameter: 56 nm•At room temperature

k = 26 W/m-°C

•Five times smaller than bulk value•At 20 K

k = 0.72 W/m-°C•6000 times smaller than bulk value!

Use bulk values with caution for microscale cases!

CHAPTER 11

MICROSCALE CONDUCTION

11.1 Introduction

11.1.1 Categories of Microscale Phenomena

• Classical Fourier law of diffusive heat conduction breaks down for• processes that are too fast

• systems that are too small

• Chapter focus is on small systems at steady-state• Essential questions

• What are the physical mechanisms by which the classical Fourier’s law will fail for small systems?

• At what length scales does this happen?

• How can we modify 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 and classical size effect

= wavepacket wavelength

L = characteristic length

= mean free path

collision

Characteristic length, LSmaller L

L ~L L~ L~~

(a) (b) (c) (d)

Fig. 11.1

•Wavepacket (Fig. 11.1.a)•A wavelike disturbance localized within a small volume of space•Particle-like

•Classical size effect (Fig. 11.1.c) •When > L >> , boundary collisions increase, impeding energy flow•Particle approximation still holds

•When << << L, wavepacket is treated as a particle (Fig. 11.1.b)

11.1.2 Purpose and Scope of this Chapter• Steady-state heat conduction at the microscale

• Key concepts of the classical size effect

• Supporting subjects not covered• Solid state physics

• Quantum mechanics

• Statistical thermodynamics

11.2 Understanding the Essential Physics of Thermal Conductivity Using the Kinetic Theory of Gases• Kinetic theory offers

• Maximum physical insight for minimal complexity

• Applicable to a wide range of realistic problems

11.2.1 Derivation of Fourier’s Law and an Expression for the Thermal Conductivity

• Energy is exchanged when particles collide

• “Mean free path” is the average of the distances a particle travels between collisions• = mean free path

• “Mean free time” is the corresponding time between collisions• = mean free time

2

1

2'

1'

Collision

Mean Free Path ()

Velocity v

2

1

(a) (b) (c)

x0 x0+x0

"left" control volume "right" control volume

(hotter) (colder)

(d)

Fig. 11.2.a through 11.2.c

v (11.1)

Derivation of Fourier’s Law by net energy flow evaluation

• Consider two adjacent control volumes of gas each with thickness and area A, subject to a temperature gradient, Fig. 11.2.d

2

1

2'

1'

Collision

Mean Free Path ()

Velocity v

2

1

(a) (b) (c)

x0 x0+x0

"left" control volume "right" control volume

(hotter) (colder)

(d)

Fig. 11.2.d

•After waiting a period = /v, half the particles in each control volume have exited through the boundaries•Particles from the left are hotter than particles from the right

ESTIMATION OF ENERGY EXCHANGETotal energy in the left control volume:

avgleftleft AUU ,ˆ

Total energy in the right control volume:

avgrightright AUU ,ˆ

Note that Uavg is the internal energy per unit volume, J/m3

The net energy crossing x = x0 is

avgrightavgleftrightleftLR UUAUUU ,,2

1ˆ2

1ˆ2

1ˆ

Assuming U is a smoothly-varying function of x that is approximated as a straight line over distance :

21

0, xUU avgleft

21

0, xUU avgright

Using a Taylor series expansion:

Assumption of local thermodynamic equilibrium:• energy density of the particles conforms to the local temperature

Exact analysis shows that the correct expression is:

By the chain rule of calculus:

dx

dTC

dx

dT

dT

dU

dx

dU

0

,,x

avgrightavgleft dx

dUUU

(11.2)

0

3

2,,

xavgrightavgleft dx

dUUU

where C is the specific heat capacity at constant volume per unit volume

vcC

Collecting results, dividing by the area and elapsed time yields Fourier’s law of heat conduction:

Therefore, the thermal conductivity of a gas of particles is:

dx

dTCv

A

Uq LR

3

1ˆ

(11.3)

Cvk 31 (11.4)

11.3 Energy Carriers• Section focus: Identify the particles in a heat-conducting

material that carry the energy• Determine the following properties: C, v, • Ideal gas, metal, insulators and heat transfer by radiation

will be investigated

11.3.1 Ideal Gases: Heat is Conducted by Gas Molecules

• Recall the ideal gas lawRTmpV tot (11.5)

wherep = absolute pressure

V = volume

mtot = total mass of the gas

T = absolute temperature in Kelvin

R = gas constant for the gas being studied, found from:

MRR U / (11.6)where

RU = universal gas constant, 8.314 J/mol-K

M = molecular weight of the gas

• Also recall that the universal gas constant can be expressed as:

BAU kNR (11.7)where

NA = Avogadro’s number, 6.022 x 1023 mol-1

kB = Boltzmann’s constant, 1.381 x 10-23 J/K

Use absolute temperature units throughout this chapter

Properties of monoatomic gases• Specific heat

• Specific heats at constant volume (cV) and constant pressure (cp) are related by:

CJ/kg Rcc Vp

CJ/m3 Tpcc Vp / (11.8)

• On a per-unit-volume basis they are related by:

T

pCV 2

3 (11.9)

• Speed• Molecular speeds are distributed over a broad range of

values: the Maxwellian velocity distribution

• This distribution is temperature dependent

• Thermal velocity is represented by the root-mean-square velocity:

22

1

dcoll

• The specific heat has the form:

RTvv rmsth 3 (11.10)

• Mean Free Path• Mean free path between collisions is:

is the average number of molecules per unit volume

d = effective diameter of the gas molecule

TkpMN BA // where

• Not all energy is exchanged during molecule collisions

• The mean free path for energy exchange is:

22 35

12

35

12

dp

Tk

dB

en

(11.11)

Note that en is about 3.5 times larger than coll, and en is the correct choice for evaluating thermal conductivity.

Example 11.1: Thermal Conductivity of an Ideal Gas

Calculate the thermal conductivity for helium at 0°C and atmospheric pressure and compare with experimental value from Appendix D.

Molecular diameter of He: d = 0.2193 nm

(1) Observations•Helium is a monoatomic gas•Kinetic gas theory applies

(2) Formulation(i) Assumptions

(1) Helium can be modeled as an ideal monoatomic gas(2) atoms are treated as elastic spheres

(ii) Governing EquationsKinetic theory expression, (11.4)

Expression for C, (11.9)

Expression for v, (11.10)

Expression for , (11.11)

Cvk 31 (a)

T

pCV 2

3 (b)

RTvth 3 (c)

235

12

dp

TkBen

(d)

(3) Solution

Specific heat:•Substituting T = 273.15 K and p = 101,300 pa into (b) yields:

Cm

J

K

Pa3

3.55615.273

300,101

2

3

2

3

T

pCV

Speed:•Atomic weight of He: M = 4.003 g/mol, so

Note that 1 Pa = 1 J/m3

KJ/kg 2077/MRR U

•Substituting R and T into (c) yields:

m/sKKJ/kg 103515.273207733 RTvth

Mean free path:•Substituting numeric values into (d) yields:

nm

mPa

J

mPa

KJ/K

260510605

102193.0300101

15.27310381.1

35

12

35

12

9

29

23

2

en

Ben

,dp

Tk

Thermal conductivity:•Substituting the values for Cv, v and into (a) yields:

Cm

W

nms

m

Cm

J3

1464.0

60513053.55631

31

k

Cvk

Comparison with experimental value:•Experimental value for thermal conductivity for He at 0°C from Appendix D: 0.142 W/m-°C

%1.3142.0

142.01464.0

error

(4) CheckingDimensional Check•Units of Cv are (J-m-3-°C-1)(m-s-1)(m), which gives (W/m-°C) – the correct units for k

(5) Comments(i) Observed 3.1% error in k is only slightly larger than the

2% uncertainty stated in (11.11) for and reflects the experimental uncertainty in d and k

(ii) Since helium’s diameter is smaller than that of all other gases, the value of is a relatively large value for an ideal gas• Molecules with larger diameters have smaller

(iii) Helium gas used to conduct heat between two parallel plates with a gap less than the mean free path will be discussed later in the chapter• This refers to “classical size effect”

11.3.2 Metals: Heat is Conducted by Electrons• Metals have high concentrations of “free electrons”

that are• Responsible for high thermal conductivity

• Dominate transporters of heat

• Theoretical results are presented

Properties of electrons in metals

• Speed• Electrons travel at the same speed, known as the Fermi

velocity vF

• Typical values of vF are around 1-2 x 106 m/s

• Free electrons are also characterized by their Fermi Energy EF, related to vF by

221

FeF vmE where me is the mass of an electron: 9.110 x 10-31 kg

• Fermi velocity is related to the concentration of free electrons e by

where is the reduced Planck’s constantsJ 3410055.1

3/123 ee

F mv

(11.12)

• Specific heat• Simple form, expressed in three ways:

TC

TTkC

ETkC

e

FBee

FBee

/

/2

21

2221

(11.13)

• Second form of (11.13) defines a characteristic “Fermi Temperature”

BFF kET / (11.14)

• Third form of (11.13) shows specific heat is proportional to temperature

• Coefficient has units of J/m3-K2

• Though electrons dominate thermal conductivity, Ce is typically several orders of magnitude smaller than reference values of C for metals

• Mean free path• Typically on the order of tens of nm at room temperature

• Free electron properties of selected metals (Table 11.1)

11.3.3 Electrical Insulators and Semiconductors: Heat is Conducted by Phonons (Sound Waves)• Insulators (dielectrics) have extreme scarcity of

electrons as compared to metals• Atomic vibrations store thermal energy

• Sound waves are the simplest class of atomic vibrations

• Wavelengths of sound waves are much larger than lattice spacing between atoms

• Wavelength and oscillation frequency follow the simple relationship

sv2

(11.15)

A “phonon” is the quantum of a sound wave in the same way that a “photon” is the quantum of a light wave

where vs is the speed of sound in the material

• There are two classes of phonons: acoustic and optical• Acoustic phonons are sound waves that follow (11.15)

with an upper limit on the allowed frequencies• At sufficiently high frequencies, the wavelengths become

comparable to the lattice constant, and it is unphysical to speak of wavelengths shorter than twice the interatomic spacing

• Optical phonons are present if and only if a material’s crystal structure has more than one atom per “primitive unit cell”

• The velocity of optical phonons is commonly set to zero, implying that optical phonons make negligible contribution to heat transfer

• Acoustic phonons will be the exclusive focus with regard to thermal conductivity

Properties of Acoustic Phonons

• The approximations used here are the Debye approximations

• Speed• Acoustic phonons are approximated by traveling at the

speed of sound, vs, in the material

• This is found by averaging the transverse and longitudinal sound speeds:

2/12,3

12,3

2 LsTss vvv (11.16)

• The above ensures that the result becomes exact at low temperature

• Specific heat• Approximation to exact Debye calculation:

3

4541

3

T

kC

D

BPUC

(11.17)

where PUC is the number of primitive cell units and D is the “Debye temperature:”

3/126 PUCB

sD k

v (11.18)

• Equation (11.17) gives less than 12% error compared to the exact Debye calculation at intermediate temperatures

• A slightly different definition of D substitutes PUC with the number density of atoms atoms

• In the limits of high and low temperature, eq. (11.17) reduces to the well-known limiting expressions

DD

BPUC TT

kC

203

3

512 4

(11.19)

DBPUC TkC 213 (11.20)

• In equation (11.19), low temperature result, C is proportional to T3 : the “Debye T3 law”

• In equation (11.20), high temperature result, C approaches a constant: the “Law of Dulong and Petit”

• An “Einstein model” is used for specific heat for optical phonons

• Handbook values for C are for total specific heat which includes contributions from both acoustic and optical phonons

• Acoustic phonon properties of selected solids (Table 11.2)

• Mean free path• Like electrons, phonon mean free path in many insulators

is approximately proportional to T-1

• At room temperature and above, thermal resistance is dominated by phonons scattering with other phonons

• Alloy atoms can also result in strong phonon scattering• Example: Ge atoms in a crystal with composition Si0.9Ge0.1

• Dopant atoms in a “doped” crystal can scatter phonons

• At temperatures around 300 K and below, effects of phonon scattering off of impurities, isotopes, defects and grain boundaries may also need to be considered

• Phonons can also scatter off sample boundaries• Classical size effect

• Discussed later in the chapter

Example 11.2: Thermal Conductivity Trend with Temperature for Silicon

Use data from Table 11.2 to propose a power law approximation of the form k(T) = aT b, where a and b are constants for thermal conductivity of bulk silicon

Assume mean free path proportional to T-1

Temperature range is 300K to 1000K

(1) Observations

•Thermal conductivity of silicon has been well-studied over a broad range of temperatures•Example limits us to the information in Table 11.2


(1) Thermal conductivity is dominated by acoustic phonons, so Debye model is adequate(2) Specific heat can be approximated by the high-T limit since the temperatures of interest are greater than D/2(3) The mean free path is assumed to vary inversely proportional to temperature

(ii) Governing EquationsSpecific heat of acoustic phonons, eq. (11.20)

DBPUC TkC 213

(3) SolutionSpeed: From Table 11.2: m/s5880sv

Specific heat:•Using values from Table 11.2 in (11.20) yields:

CJ/m

J/Km3

6

23328

1004.1

10381.1105.233

C

kC BPUC

Thermal conductivity at 300 K:•From Table 11.2:

•C is assumed constant from 300 K to 1000 K

C-W/m 148k

Mean free path:•Combining values for k, C and v at 300K:

nm

m/sCJ/m

CW/m3

7358801004.1

148336

Cv

k

•Since varies in proportion to T -1:

KK 300300

.

.

T

constTT

const

Thermal conductivity power law:•Consider k(T)/k(300K)•From kinetic theory:

KK 300300 3

1

31

Cv

TCv

k

Tk

•Since C is approximately constant:

KK 300300 T

k

Tk

•Using the result for :

1300148

300300

300

300

T

TkTk

Tk

Tk

KKW/m

KK

K

K

•The final result is:

1400,44 TTk W/m

•Comparing to the power law form:

1

440,44

b

a W/m

where T must be expressed in Kelvin

(4) CheckingDimensional Check•Right hand side of last eq. has units of K-1-W/m which is equivalent to expected units of W/m-°C

Magnitude Check•Calculations are compared to standard reference values from Appendix D:

(5) Comments(i) Mean free path is best measured in nm; values in the

range of tens to hundreds of nm are typical for phonons in dielectric crystals at room temperature

(ii) Using density of silicon = 2330 kg/m3 from Appendix D, acoustic specific heat is converted to a mass basis, yielding 446 J/kg-°C• Handbook value in Appendix D is 712 J/kg-°C• Optical phonons make a significant contribution to

the total specific heat

(iii) The T-1 power law is only approximate• Actual thermal conductivity varies by a factor of 4.74,

compared to the expected variation by a factor of 3.33• A better power law in this range is

3.1Tk

(iv) Power law is strongly temperature dependent•Below 10 K, power law is

3Tk

Example 11.3: Bulk Mean Free Paths as a Function of Temperature

Estimate L for acoustic phonons in bulk silicon as a function of temperature

Temperature range is 1 K to 1000 K

Use approximate Debye model for specific heat and sound velocity

Thermal conductivity of bulk silicon from Appendix D

(1) Observations•Temperature range is both far above and far below D

•Above room temperature, phonon-phonon scattering dominates the mean free path with an approximate power law of 3.11 TT or

•At low temperature, thermal conductivity climbs as T is reduced from 300 K to 20 K, but then falls rapidly as T is further reduced


(1) Thermal conductivity is dominated by acoustic phonons over the entire range(2) The Debye model is adequate

(ii) Governing EquationsSpecific heat eq. (11.17) will be used

3

4541

3

T

kC

D

BPUC

Sound velocity is taken from Table 11.2

(3) SolutionMean free path is given by

Cvk /3From equation (11.17)

3

4541

3

T

kC

D

BPUC

where from Table 2:

K

m

512

105.2 328

D

PUC

m/s5880sv

Mean free path is calculated by combining the above with the data for k from Appendix D.

Tabulated and graphical results:

Fig. 11.3

(4) CheckingLimiting Behavior Check•The specific heat transitions from T 3 at low temperature to constant at high temperature, as expected•Transition occurs at 120 K, which is near the expected transition of D/3 = 170 K

•Trends of both mean free path and thermal conductivity are approximately T -1 at high temperature, as expected

Value Check•At 300 K, C is 9.73 x 105 J/m3-°C, which is close to 1.04 x 106 J/m-°C found in Ex. 11.2 using high-temperature approximation• is 78 nm, which is close to 73 nm found in Ex. 11.2

(5) Comments

Low-temperature behavior is distinctive• Mean free path appears to saturate at 4 nm• Since low-temperature specific heat goes as T 3, so

does the low-temperature thermal conductivity• Cubic trend is general for phonon thermal

conductivity at low temperature• Behavior is dominated by classical size effect at

low temperature• of 4 nm corresponds to characteristic length of

sample used to generate values of k in Appendix D• Other reference values for bulk silicon at low

temperatures may differ due to sample size, but high temperature values should correlate well

11.3.4 Radiation: Heat is Carried by Photons (Light Waves)• Using kinetic theory, radiation and conduction are

seen as two limiting cases of a single phenomenon• Radiation is treated as a gas of photons• Speed

• Speed of light in vacuum is c = 2.998 x 108 m/s

• Specific heat• Photons store energy similarly to molecules, electrons

and phonons• Assuming perfect vacuum conditions, specific heat of a “gas”

of photons iscTC /16 3 (11.21)

where is the Stephan-Boltzmann constant, 5.670 x 10-8 W/m2-K4

• Mean free path• Varies tremendously

• Much longer than that of molecules, electrons and phonons

• Depends on physical situation

• Sun surface to earth: 90 million miles without scattering

• Vacuum chamber: mean free path is on order of 10-100 mm

• Crystals transparent in infrared (i.e. glasses): mean free path is dependent on wavelength and material, magnitude is microns to mm

Example 11.4: Effective Thermal Conductivity for Radiation Heat Transfer Between Two Parallel

Plates (One Black, One Gray)

Net radiation heat transfer from a gray plate “1” to a parallel black plate “2” is: 424

1 TTAQ (11.22)

where is the emissivity of plate 1

the plates have the same area A

the gap L is smaller than the length and width

Use kinetic theory to re-express heat transfer in terms of a “conduction thermal resistance” R = L/kradA where krad is the effective thermal conductivity of a photon gas

Derive effective mean free path

Assume temp. differences are smaller than average temp.

Evaluate for A = 0.1m2, L = 0.001m, = 0.2, T1 = 600 K, T2 = 500 K

(1) Observations•Since there is an expression for the speed and specific heat of photons, it should be possible to express radiation as conduction•In a vacuum, there are no scattering mechanisms other than the plates themselves, so the photon mean free path should be proportional to L


(1) Small temperature differences:

2121

21 TTTT (2) Properties such as specific heat can be evaluated at the average temperature:

2121 TTT

(ii) Governing EquationsRadiation heat transfer equation (11.22):

4241 TTAQ

Kinetic theory equation (11.4):

Specific heat of photons (11.21):

Cvk 31

cTC /16 3

(3) SolutionConduction resistanceEquation (11.22) is to be rearranged into the form:

RQTT 21

where R is the desired conduction resistance

Define the temperature difference:

21 TT Therefore:

21

221

1 TTTT and

Substituting the above into (11.22):

4214

21 TTAQ

Factoring out T 4:

42

4

24 11 TTATQ

From the binomial theorem or Taylor series expansion:

111 xnxx n ifSo, since << T:

TT 24

2 11

Then TTATQ 224 11

Simplifying:34 ATQ

which is rearranged:

321 4 AT

QTT

The conduction resistance is therefore:

321

4

1

ATQ

TTR

(11.23)

Effective thermal conductivityComparing (11.23) to the standard form R = L/kA, solve for k:

LTkAk

L

AT radrad

33

44

1

Effective mean free pathComparing krad to the standard form (11.4), solve for eff :

Cv

LT

CvLTk

eff

effrad

3

313

12

4

Substituting (11.21) for C, and recognizing v = c:

LvcT

LTeff

43

3

3

/16

12 (11.24)

Numerical calculationFrom the exact radiation equation, Q = 76.1 WFrom conduction resistance equation with T = 550K,R = 1.33 K/W, resulting in Q = 75.5 WThis is within 1% of the exact value

(4) CheckingDimensional Check•Conduction resistance has units of[(K4-m2-W-1)(m-2)(K-3)], yielding the correct units of K/W•Effective mean free path has units of length

Trend Check: conduction resistance•Equivalent conduction resistance is inversely proportional to area

•Reducing the area reduces heat transfer, thus increasing thermal resistance

•Conduction resistance is inversely proportional to emissivity

•Reducing emissivity should reduce heat transfer in equal proportion

Trend Check: mean free path•Effective mean free path is proportional to L

(5) Comments(i) Classic radiation heat transfer can be viewed as heat

conduction if krad is defined appropriately(ii) Effective thermal conductivity of a photon gas is

proportional to temperature cubed• Higher energy results in many more photons

(iii) Effective mean free path captures emissivity effects• Shinier plates have shorter mean free paths

(iv) Thermal conductivity krad can vary on orders of magnitude depending on the temperature and geometry

(v) The result that eff = ¾ L for = 1 will appear again when discussing heat conduction in thin films

(vi) If the vacuum between the plates is filled by a dense gas or other medium that significantly scatters the photons, eff and the radiation heat transfer will be reduced

11.4 Thermal Conductivity Reduction by Boundary Scattering: The Classical Size Effect

• Mean free path for “traditional applications” is several orders of magnitude smaller than sample size• Sample sizes range from millimeters to meters• Handbook values for thermal conductivity apply

• For technologies where the characteristic length is on the order of 10-100 nm, energy carriers collide frequently with structure boundaries• This is the classical size effect

11.4.1 Accounting for Multiple Scattering Mechanisms: Matthiessen’s Rule• For cases where classical size effect is important,

thermal conductivity is still expressed as

effCvk 31

where eff is the mean free path

• Speed and specific heat of energy carriers in nanostructures will be approximated as identical to bulk sample values:

bulkeffbulkeff vvCC and

• Efforts are fully focused on determining eff, where

bulkeff

• Matthiessen’s rule• The effective mean free path is the sum of the mean free

paths corresponding to each of the scattering mechanisms, summed in the reciprocal sense

• Provides excellent physical insight

• Mathematically, it is:

iieff

mechanism

11 (11.25)

• Example of Matthiessen’s rule: Phonon scattering• Phonons scatter on impurities, on grain boundaries, on

electrons, on other phonons and on the sample boundaries

• Five terms in sum of equation (11.25):1111

..1.

1

1

bdyphphephbgimpeff

bulk

• As in example, scattering mechanisms are partitioned into two simple categories:• Mechanisms present in a large bulk sample

• Additional mechanisms due to “boundary scattering” present in small samples only

• Matthiessen’s rule is thus expressed as:

bdybulk

bdybulkbdybulkeff

111 (11.26)

• Two cases of heat transfer in nanostructures are discussed• Heat transport parallel to boundaries, i.e. along the

plane of a thin film

• Heat transport perpendicular to boundaries, i.e. perpendicular to a thin film

11.4.2 Boundary Scattering for Heat Flow Parallel to Boundaries• Standard configurations for heat flow parallel to

boundaries (Fig. 11.4.a)

Nanowires• Mean free path for nanowire of diameter D due to

boundary scattering is

p

pDbdy 1

1 (11.27)

where p is the specularity

Thin film

Superlattice

Nanowire

(a) Heat flow parallelto boundaries / interfaces

(b) Heat flow perpendicularto boundaries / interfaces

Thin filmSuperlattice

Fig. 11.4.a

• Specularity is defined (Fig. 11.5):

• p = 0: scattering is 100% diffuse (rough surfaces)

• p = 1: scattering is 100% specular (smooth surfaces)

• 0 < p < 1: scattering p is specular and (1 – p) diffuse

Incident Reflected

Boundary

IncidentReflected

Boundary(b)(a)

a) Diffuse reflection b) Specular reflection

Fig. 11.5

• Specularity and surface roughness

• Surface roughness < wavelength: specular reflections

• Surface roughness > wavelength: diffuse reflections

2

2316exp

p (11.28)

• Intermediate surface roughness is estimated using

where is the root mean square roughness

is the wavelength of the energy carrier

Thin films (Diffuse: p = 0)• For a thin film of thickness L, standard result for mean

free path is the Fuchs-Sondheimer solution:

bulkbulk

bulk

bulk

eff LE

LE

L

53 4418

31 (11.29)

where E3 and E5 are “exponential integrals” defined using

1

0

2 /exp dxxE nn

where is a dummy variable of integration

• Exponential integrals E3 and E5 are shown in Fig. 11.6

• These functions are also tabulated, and available as functions in mathematical software packages

0

0.25

0.5

0 1 2 3

3E

5E

53 EE

x

Fig. 11.6

• For thick and thin films, the following asymptotic forms are useful:

bulkbulk

bulk

eff LL

1

8

31 (11.30a)

bulk

bulk

bulk

bulk

eff LLL

4

/ln3(11.30b)

• Equation (11.30a) compared with exact solution:

• Errors less than 10% for L > 0.5bulk

• Errors less than 20% for L as small as 0.025bulk

• Equation (11.30b) compared with exact solution:

• Errors less than 10% for L < 0.019bulk

• Errors less than 20% for L as large as 0.125bulk

Thin films (Some specularity: p > 0)

• Effective thermal conductivity increases when specularity becomes significant

• Mean free path is given by:

1

0

3

/exp1

/exp1

2

131

d

Lp

L

L

p

bulk

bulkbulk

bulk

eff

(11.31)

• Equation (11.31) is readily evaluated numerically

• Figure 11.7 shows resulting eff for a range of specularities

• p = 0 case reduces to (11.29)

• In the p = 1 case there is no reduction of k compared to bulk value, regardless of L

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10 100

L / bulk

ef

f /

bulk =

kef

f / k

bulk p=0.99

p=0.9

p=0

p=0.2

p=0.5

Fig. 11.7

Example 11.5: Thermal conductivity of a silicon nanowire

Find the specularity, p, and the effective thermal conductivity, k, at T = 300K for a silicon nanowire with diameter D = 56 nm

Assume the surface roughness is approximately = 0.5 nm

Assume the average phonon wavelength at 300 K is = 1 nm

(1) Observations•Roughness is slightly smaller than wavelength, so specularity is expected to be in the transition region:0 < p < 1•From Table 11.4, mean free path of acoustic phonons in bulk silicon from Debye approximation is

nmK 78300 bulk

•Since nanowire diameter is comparable to bulk, thermal conductivity is expected to be significantly reduced as compared to bulk value


(1) Thermal conductivity is dominated by acoustic phonons, so Debye model is adequate(2) Specific heat and phonon speed in the nanowire are identical to those in bulk silicon

(ii) Governing equationsSpecularity is calculated from (11.28)

2

2316exp

p

Boundary scattering mean free path (11.27)

p

pDbdy 1

1

This is so small that the specularity is effectively zero

bdybulk

bdybulkbdybulkeff

111

(3) SolutionSpecularityFrom (11.28), for / = 0.5:

54

232

23

1037.1124exp

5.016exp16

exp

p

p

Matthiessen’s rule (11.26) for effective mean free path

Boundary scattering mean free pathFrom (11.27), with p = 0:

nm5601

01

1

1

DDp

pDbdy

Effective mean free pathFrom Matthiessen’s Rule (11.26):

nm

nmnm

nmnm33

5678

5678

bdybulk

bdybulkeff

Thermal conductivityConsider the ratio k/kbulk:

bulk

eff

bulk Cv

Cv

k

k

31

31

Since C and v are independent of structure size:

bdybulkbdybulk

bdy

bulk

eff

bulkk

k

/1

1

or, using bdy = D:

Dk

k

bulkbulk /1

1

(11.32)

ThereforeCW/mor 62,42.0/ kkk bulk

(4) CheckingDimensional Check•Since all equations are dimensionless ratios, there are no dimensions to check separately

Limiting Behaviors•When the diameter is much larger than the bulk mean free path, eq. (11.32) reduces to k = kbulk, showing diameter no longer matters in this case•In opposite limit, when D << bulk,

bulkbulk Dkk /which can be written as:

CvDk 31

so further reductions in nanowire diameter should reduce the thermal conductivity in direct proportion

(5) Comments(i) Thermal conductivity for a nanowire like this one has

been measured at 300 K, and found to be 25.7 W/m-°C• Debye estimate of bulk (78 nm) is too conservative

(ii) A better estimate for bulk of Si at 300 K is 200-300 nm• This accounts for the frequency dependence of C

and v contributions from various phonons• When bulk = 250 nm is used, k = 27 W/m-°C, which

agrees well with experimental value

(iii) Debye estimate for mean free path is commonly too low by a factor of 2• Born-von Karman approach gives better

approximation of bulk = 210 nm• Using this, k = 31 W/m-°C• This requires numerical integration to solve for

Cv so is not pursued here

Example 11.6: Temperature Dependence of the Thermal Conductivity of a Nanowire

Plot thermal conductivity as a function of temperature for Si nanowires of several diameters

Temperature range is from 1 K to 1000 K

Use approximate Debye model for specific heat

Use bulk mean free path as a function of temperature from Example 11.2

Assume purely diffuse scattering

Diameters: Macro-wire (D = 1 mm)

Micro-wire (D = 1 m)

Nanowire (D = 56 nm)

Compare calculations for the 56 nm nanowire with experimental measurements found in Appendix D

(1) Observations•Based on results from Ex. 11.3, it is known that boundary scattering dominates thermal conductivity at low temperature•Boundary scattering becomes stronger for small samples, so the thermal conductivity is expected to be reduced as D is reduces


(1) Thermal conductivity is dominated by acoustic phonons so Debye model is appropriate(2) Specific heat and phonon speed in the nanowire are identical to the bulk values(3) Boundary scattering is perfectly diffuse (p = 0)

(ii) Governing EquationsEquation (11.17) is used for specific heat

bdy is calculated with eq. (11.27)

Matthiessen’s Rule (11.26) is used for eff:

Sound velocity is taken from Table 11.2

Bulk mean free path is taken from Table 11.4

3

4541

3

T

kC

D

BPUC

p

pDbdy 1

1

bdybulk

bdybulkbdybulkeff

111

(3) SolutionFrom equation (11.27) with p = 0, bdy = D as in Example 11.5Bulk mean free paths were calculated in Ex. 11.3, so Matthiessen’s Rule is used to calculate the effective mean free path:

1

1

DD

D bulkbulk

bulk

bulk

bdybulk

bdybulkeff

Multiplying both sides by 1/3 Cv:1

1

Dkk bulk

bulkeff

Resulting mean free paths and thermal conductivities, Fig. 11.8

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1 10 100 1000

Temperature [K]

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1 10 100 1000

Temperature [K]

eff (D=1 mm)

eff (D=1 micron)

D=56 nm, experiments(D. Li et al.)

bulk (from Table 11.4)

Bulk

D=1

mm

D=1

mic

ron

D=5

6nm

D=56 nm, improved calculation.

eff (D=56 nm)

Me

an F

ree

Pa

th [m

]T

herm

al C

ondu

ctiv

ity [

W/m

-C

]

Fig. 11.8

At 300 K: thermal conductivities are about 5 times different

At 20 K: thermal conductivities are over 6000x different

(4) CheckingTrend Check•Calculated thermal conductivities of the nanowires have the same general shape as a function of temperature as for bulk Si

•Increase as T 3 at low temperature•Reach a peak value•Decrease at high temperature

•Effective mean free paths become constant at low temperature and decrease at high temperature•Reducing the diameter leads to lower thermal conductivity•At very low temperature

Dk as expected

(5) Comments(i) Diameter effect is much more dramatic at low T than

at high T• Bulk mean free paths are short at high T, so boundary

scattering is less important

(ii) Reducing the diameter reduces the peak value of the k(T) curve and shifts peak to higher temp

(iii) Limiting behavior for smaller diameters• For diameter smaller than the bulk mean free path,

boundary scattering dominates over entire temperature range and k(T) curve reproduces shape of C(T) curve

(iv) Limiting behavior for infinitely large diameters• For infinitely large diameters, thermal conductivity curve

diverges at low temperatures because scattering mechanisms weaken

(v) Debye model explains all the major trends found in experimental results, but model disagrees with experiments by a factor of 2 at high temperatures.

11.4.3 Boundary Scattering for Heat Flow Perpendicular to Boundaries• Standard configurations for heat flow

perpendicular to boundaries (Fig. 11.4.b)Thin film

Superlattice

Nanowire

(a) Heat flow parallelto boundaries / interfaces

(b) Heat flow perpendicularto boundaries / interfaces

Thin filmSuperlattice

Fig. 11.4.b

• Kinetic theory is in general less appropriate for these configurations

• Still gives physical insight that is accurate for simple cases

Thin films with no heat generation• For thin film of thickness L without heat generation,

kinetic theory and Matthiessen’s Rule can be used if boundary scattering mean free path is expressed by:

112

11

43

Lbdy (11.33)

where 1 and 2 are the absorptivities of the two bounding surfaces

• Concept of emissivity and absorptivity is generalized from radiation to other types of energy carriers

• In context of heat conduction by gases, absorptivities are replaced by “energy accommodation coefficients”

• Assume emissivity and absorptivity are approximately equal (Kirchoff’s Law)

• Eq. (11.33) is an expression of Rosseland diffusion approximation

• Deissler jump boundary conditions

• Is derived in field of radiation heat transfer in a participating medium

• Is also derived using the Boltzmann transport equation

• This is a simplified result in the form of mean free path for boundary scattering

Example 11.7: Thermal Conductivity Perpendicular to a Silicon Thin Film

Silicon thin film of thickness L at 300 K sandwiched between two heat sinks

Assume phonon absorptivities of the two contacts are perfect

Plot the effective thermal conductivity as a function of film thickness over L range from 1 nm to 1 mm

Plot the conduction thermal resistance for a sample of 1 cm2 cross-sectional area

(1) Observations•From Table 11.4, bulk = 78 nm from Debye approximation at 300K, though Ex. 11.5 suggestsbulk = 250 nm is a better choice

•Because the thinnest films (L = 10 nm) are much thinner than bulk, k << kbulk is expected


(1) Thermal conductivity is dominated by acoustic phonons so Debye model is appropriate(2) Specific heat and phonon speed are identical that in the bulk material(3) Phonon absorptivities and emissivities are approximated as unity at both contacts; no other contact resistances are considered(4) Bulk mean free path at 300 K is bulk = 250 nm

(ii) Governing EquationsEquation (11.17) is used for specific heat

3

4541

3

T

kC

D

BPUC

bdy is calculated with eq. (11.33)

Matthiessen’s Rule (11.26) is used for eff:

bdybulk

bdybulkeff

112

11

43

Lbdy

(3) SolutionSpecific heatSame as bulkUsing (11.17) with D = 512 K and PUC = 2.50 x 1028 m-3 from Table 11.2, at 300K:

SpeedSame as bulk: v = 5880 m/s

CmJ

KK

J/Km

35

45

23328

3

45

/1073.9

300512

1

10381.1105.23

1

3

44

C

T

kC

D

BPUC

Effective mean free pathFor 1 = 2 = 1, (11.33) simplifies to

LLL

bdy 434

3

12

11

43

1111

Substituting into (11.26) yields

L

L

L

bulk

bulkeff

bulk

bulk

bdybulk

bdybulkeff

34

1

43

43

Thermal conductivityCombining C, v and eff using (11.4):

which is rewritten as

bulk

bulk

bulk

bulkeff L

LCv

L

CvCvLk

43

3

41

31

31

(11.34)

Lk

Lk

bulkbulk

3

41

1

(11.35)

Thermal resistanceConduction thermal resistance is R = L/kA, so substituting from (11.34)

Ak

L

kA

LLR

bulk

bulk34 (11.36)

Plots of functional dependencies of various quantities on the film thickness L , Fig. 11.9

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.001 0.01 0.1 1 10 100

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

0.001 0.01 0.1 1 10 100Film thickness, L [micron]

1.0E-05

1.0E-04

1.0E-03

1.0E-02


1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.001 0.01 0.1 1 10 100

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03


1.0E-05

1.0E-04

1.0E-03

1.0E-02


Diffusive

Ballistic

Diffusive

Ballistic

Diffusiv

e

Ballistic

Spe

cific

Hea

t C

[J/m

3 - C

]

The

rmal

Con

duct

ivity

k[W

/m-

C]

Spe

edv

[m/s

]

Mea

n F

ree

Pat

heff

[m]

The

rmal

Res

ista

nce

R[K

/W]

1.0E+05

1.0E+06

1.0E+07


1000

10000

0.001 0.01 0.1 1 10 100

Film thickness, L [micron]

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.001 0.01 0.1 1 10 100

1.0E+05

1.0E+06

1.0E+07


1000

10000

0.001 0.01 0.1 1 10 100


1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.001 0.01 0.1 1 10 100

Diffusive

Ballistic

Diffusive

Ballistic

Diffusiv

e

Ballistic

Spe

cific

Hea

t C

[J/m

3 - C

]

The

rmal

Con

duct

ivity

k[W

/m-

C]

Spe

edv

[m/s

]

Mea

n F

ree

Pat

heff

[m]

The

rmal

Res

ista

nce

R[K

/W]

1000

10000

0.001 0.01 0.1 1 10 100


1.0E-03

1.0E-02

1.0E-01

1.0E+00


1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

0.001 0.01 0.1 1 10 100

1000

10000

0.001 0.01 0.1 1 10 100


1.0E-03

1.0E-02

1.0E-01

1.0E+00


1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

0.001 0.01 0.1 1 10 100

Diffusive

Ballistic

Diffusive

Ballistic

Diffusiv

e

Ballistic

Spe

cific

Hea

t C

[J/m

3 - C

]

The

rmal

Con

duct

ivity

k[W

/m-

C]

Spe

edv

[m/s

]

Mea

n F

ree

Pat

heff

[m]

The

rmal

Res

ista

nce

R[K

/W]

Fig. 11.9

effCvk 31

C and v: independent of L

MFP: Matthiessen’s rule

(4) CheckingDimensional check•All terms in (11.35) are dimensionless•In (11.36), numerator terms have units of (m) and denominator has units of (W/m-°C)(m2), giving units of resistance (K/W)

Limiting behavior•In the limit L >> bulk:

•Eq. (11.35) correctly reduces to k = kbulk = 148 W/m-°C

•Eq. (11.36) correctly reduces to R = L/kbulkA

(5) Comments(i) The sequence of plots in Fig. 11.9 summarizes classical

size effect• The qualitative behavior and trends are general

(ii) By eliminating Cv from eq. (11.35) and (11.36), the results for k(L) and R(L) are more accurate

(iii) Thick film limit, L >> bulk, is known as the “diffusive” regime• Thermal resistance is dominated by diffusion of energy

carriers from hot side to cold side• Here, thermal resistance is linearly proportional to L

(iv) For limiting behavior for thin film, L << bulk, “ballistic” regime• k is proportional to L• R becomes independent of L• Identical to traditional black body radiation resistance

(v) From Matthiessen’s Rule, transition occurs when bulk is comparable to bdy

Table 11.5: Comparison between diffusive and ballistic limits

11.5 Closing Thoughts• Kinetic theory framework was used to introduce

essential concepts of conduction heat transfer by energy carriers in bulk and nanostructures

• For any energy carrier and nanostructure, key tasks are determination of• Specific heat C• Carrier velocity v• Effective mean free path eff

• Problems were limited to classical size effect• Kinetic theory can be generalized to account for

frequency dependence of properties• Eq. (11.4) is written as an integral over frequency

dTvTCTk eff ,,)( 31 (11.37)

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