Thermal Wave Model

The original thought of the thermal wave theory is that the speed of heat propagation cannot be infinity. Thus, there must be a time lag between heat flux and temperature gradient [24]. Cattaneo [25] and Vernotte [26] proposed a modified model as following:

T and T is the temperature. Substitution the above equation into the energy conservation equation, then an alternative heat transfer model which is known as thermal wave model or hyperbolic heat conduction equation was developed. The hyperbolic heat conduction equation can be expressed as

T T

R T 2

2 + 1 ∂ =∇

∂

τ , (1-5)

and phonon can be described as a two-step process [27-29].

where C is the heat capacity, k is the thermal conductivity, and the subscript indices e and p denoting for electron and phonon, respectively. The energy exchange between electrons and phonons is characterized by the coupling factor G .

1.3.4 Phonon Scattering Model

The phonon scattering model was developed by solving the linearized Boltzmann transport equation for the pure phonon field [30]. Only heat transport by phonon scattering was emphasized and the contribution of electron in conducting heat was neglected. and τ is the relaxation time of elastically scattering. _N

becomes

Where f is the equilibrium phonon distribution, which follows the Bose-Einstein ⁰ distribution. Majumdar transformed the Boltzmann equation to an equation of phonon radiative transfer (EPRT) in the form [31]



where I is the phonon intensity, and µ is the direction cosine. This equation has the same form as the equation of radiative transfer. Therefore, Majumdar called it the equation of phonon radiative transfer.

1.4 Interface Thermal Resistance Models

1.4.1 Interfacial Layer Model

The interfacial layer model assumes an interfacial layer with variable thickness and thermal conductivity in between two dissimilar materials. Two very important parameters directly and significantly affecting heat transport across the two dissimilar materials are the thermal and thickness ratios of the interfacial layer to the reference material. For example, according to the experimental results of Marshall et al. [32] the thermal conductivity and thickness ratios of the interfacial layer to Y-Ba-Cu-O thin

1.4.2 Acoustic Mismatch Model

The acoustic mismatch model starts by assuming that phonons are the major carriers of heat. The interface thermal resistance is attributed to the transmitting and reflecting of phonons at the interface. The interface is perfect and structureless so there is no scattering occurring at the interface. The phonon transmission probability from medium 1 to medium 2 is governed by the Fresnel equations and can be expressed as [33]

2

where ρ is the density and v is the sound velocity. Before proceeding the AMM, the range of validity of assumptions should be checked. At very low temperatures, the phonon wavelength is much smaller than the interfacial roughness and the size of defect near the boundary. The interface may appear to be a perfectly flat plane, which is the basic assumption in the acoustic mismatch model. Thus, the AMM works well at low temperatures. For example, Herth and Weis [34] measured the interface resistance between gold film and sapphire substrate at 100K and confirmed that the AMM predictions matched the experimental results.

1.4.3 Diffuse Mismatch Model

For most interfaces above a few Kelvin, the interface itself is a very strong scatterer of phonons. The analog to the Fresnel equations is no longer valid. Instead,

The acoustic mismatch model is valid for smooth interfaces and the diffuse mismatch model is adequate for rough interfaces. The ratio of phonon wavelength to the interface roughness determines the applicability of AMM and DMM. When the phonon wavelength is far small than the interface roughness, the interface can be seen as a rough interface and the diffuse mismatch model is applicable. Otherwise, the acoustic mismatch model is more suitable for a smooth interface which roughness is small compared to the phonon wavelength.

1.4.4 Scattering -Mediated Acoustic Mismatch Model

Scattering- mediated acoustic mismatch model (SMAMM) was developed by observing the close analog between radiative and phonon heat transport [36].

Scattering near the interface is dominant than any other mechanism [35]. At high temperatures, the phonon mean free path is short, so that the phonon wave will attenuate rapidly. In contrast, scattering has hardly any effect on the transmission and reflection of acoustic waves at very low temperatures. The reflectivity from medium 1 to medium 2 predicting by SMAMM can be expressed as

2

τω

SMAMM accurately describes the behavior of interface thermal resistance at high temperatures and it can reduce to AMM at low temperatures.

1.5 Literature Survey

1.5.1 Microscale Heat Transfer

As mentioned before, Fourier law, a macroscale model, is inadequate to deal with microscale heat transfer problems. Thus, it is important to develop microscale heat transfer model. Preliminary to develop microscale heat transfer models, the thermophysical characteristics of heat carriers must be clarified. An understanding of heat carriers is the first step toward solving microscale heat transfer problems. The

boundary between the macroscale and microscale transfer regimes. Further, Majumdar [38] listed the characteristic time- and length scales and corresponding transport phenomena of energy carriers, as shown in Table 1.2. Wave transport and microscopic particle transport are suitable for describing microscale heat transfer in solids. The wave and particle nature of phonons can be obtained from classical text by Kittel [39]. Recently, Chen [40] studied the phonon wave effects on heat conduction in thin films. The results had demonstrated that it is safe to treat the phonon transport based on the phonon particle picture for thin films of practical thickness. The Boltzmann Transport Equation (BTE) is the most suitable model for describing phonon transport in solids due to its ability to correctly describe both equilibrium and non-equilibrium phenomena [41]. To solve the Boltzmann transport equation, several approximation methods had been proposed by numerous researchers [42]. According to BTE, Majumdar [31] developed an equation of phonon radiative transfer (EPRT) to deal with microscale heat conduction in dielectric thin films. Later, Joshi and Majumdar [43] employed the EPRT to study heat transfer across diamond thin films for both steady-state and transient cases. In their studies, the results demonstrated that the geometric size has a great influence on the thermal properties of materials, which is so-called size effect.

1.5.2 Interface Thermal Resistance

resistance, theoretical analysis also has been done by many researchers. Little [49]

predicted interface thermal resistance by treating phonons as plane waves and proposed the acoustic mismatch model (AMM). An essential assumption of the AMM is that no scattering occurs at the interface. Swartz and Pohl [35] considered the diffuse scattering occurring at the interface and proposed the diffuse mismatch model (DMM). Phelan [50] pointed out that the applicability of AMM and DMM is determined by the ratio, λ_d σ , where λ is the dominant phonon wavelength, and _d σ is the mean interfacial roughness. When λ_d σ >>1, the AMM is applicable, otherwise, the DMM applies. Chen [51] examined the effect of interface conditions on the thermal conductivity of superlattices in the direction parallel to the film plane. He divided the interface conditions into diffuse scattering and specular scattering. It was concluded that diffuse interface scattering, rather than specular scattering, is the cause of the observed reduction in thermal conductivity of superlattices. La ter, Chen [52]

investigated the interface effect on the thermal conductivity of superlattices in the cross-plane direction. The results based on DMM were in reasonable agreement with experimental results by Capinski and Maris on a GaAs/AlAs superlattices [53]. Zeng and Chen [54] examined the applicability of the thermal boundary resistance to the case with internal heat generation. Prasher and Phelan [36] developed a model, called the scattering- mediated acoustic mismatch model (SMAMM), to exploit the analo gy between phonon and radiative transport by developing a damped wave equation to

1.5.3 Thermoelectrics

In 1821, Seebeck discovered that when two different conductors were joined together and one of the junctions was heated, a voltage appeared. Later, Peltier discovered the reverse phenomenon that a temperature gradient appeared when an electric current passed through the junction between two conductors [56]. The discovery of thermoelectric materials offers the conversion between heat and electricity [57]. A typical thermoelectric power generator, as illustrated in Fig. 1.5, is based on the Seebeck effect. Both electrons in n-type semiconductors and holes in p-type semiconductors transport from the hot side toward the cold side due to thermal diffusion. Thus, a closed loop is formed and the current flows through an external load to do useful work [58]. Thermoelectric coolers, as shown in Fig. 1.6, is based on Peltier effect and work in reverse to thermoelectric power generators. A current flows through all the elements in series such that both electrons in n-type semiconductors and holes in p-type semiconductors leave the cold side to the hot side. During the transport process, electrons and holes carry thermal energy from the cold side to the hot side and served as heat pumps. In general, practical thermoelectric coolers are consisted of a number of n- and p-type thermoelements [59], as illustrated in Fig. 1.7.

The current flows in series through all the semiconductor thermoelements and energy flows in parallel from the cold side to the hot side.

The most important issue in thermoelectrics is to increase the figure of merit,

crystal. For bulk materials, the use of “phonon rattlers” [61] will reduce the phonon thermal conductivity. Phonon rattlers are interstitial atoms inserted into the empty space in the host material. These atoms are weakly bound by the host material [62].

Their vibrations are not consistent with atoms in the host material and thus scatter the phonon in the original lattice. As a result of phonon scattering, the lattice thermal conductivity will decrease. Another approach to reduce thermal conductivity of thermoelectric materials is to develop low-dimensional thermoelectric materials, such as nanodots (0D), nanowires (1D) and thin films (2D) [63, 64]. Low dimensionality provides several advantages: (1) one way for enhancing the density of states near the Fermi energy, resulting in an increasing Seebeck coefficient; (2) opportunities to take advantage of the anisotropic Fermi surfaces in multi- valley cubic semiconductors; and (3) opportunities to increase boundary scattering of phonons at the barrier-well interfaces, without an large increase in electron scattering at the interface [65]. An increase in boundary scattering of phonons reduces the lattice thermal conductivity of materials. Size and interface effects are utilized to develop novel thermoelectric materials. Thin film superlattices and nanowire superlattices are very popular in designing high performance thermoelectric devices.

Since many applications require materials in large quantities, it is desired to develop a bulk material with high ZT value. The highest ZT in a bulk thermoelectric material at 300K is 1.14 for p-type (Bi₂Te₃)_0.25(Sb₂Te₃)_0.72(Sb₂Se₃)_0.03

enhancement of ZT in bulk materials, the thermoelectric figure of merit of low-dimensional materials gets great improvement in recent years.

Venkatasubramanian et al. [69] reported a maximum ZT of 2.4 for the p-type Bi2Te3/Sb2Te3 thin- film superlattices devices. Up to date, it is the highest value observed by researchers.

1.5.4 Thermal Conductivity Measurements in Micro/Nano Structures

Many studies on the thermal conductivity of thin films have been carried out [1].

Besides the establishment of theoretical models, measurements of thermophysical properties are also very important. Thermal conductivity characterization of micro/nano structures is challenging since it is difficult to establish and to measure the temperature difference over a small distance [63]. Some researchers developed several measurement methods to quantify the thermal conductivity of materials. Yao [70]

measured the thermal properties of GaAs/AlAs superlattice. Yu et al. [71] reported the temperature dependence of thermophysical properties of GaAs/AlAs superlattices.

Both Yu and Yao et al. measured the thermal conductivity of superlattices in the in-plane direction. By contrast, a popular method developed by Cahill [72] for measuring the thermal conductivity of thin films in the cross-plane direction is the 3ω method. Lee and Cahill [73] measured the thermal conductivity of Si/Ge superlattices.

conductivity of an n-type Si/Ge quantum-dot superlattice in the cross-plane direction.

In the 3ω method, a metal line that was used as a heater/thermometer was deposited onto the sample, as illustrated in Fig. 1.8. Since thermoelectric thin films are electrically conducting, there is a need to deposit an electrically insulating layer to isolate the thermoelectric film from the heater. An ac current with angular frequency ω is applied to the heater and then generates a ω2 temperature variation inside the sample. Since the variation of electrical resistance of a metal is proportional to its temperature variation, the electrical resistance of the metal line oscillates at 2 . ω Multiplying the current and the electrical resistance together, and the product is the voltage drop across the heater. Thus, the voltage contains a 3 signal that depends ω on the temperature rise. Measuring the voltage change and the temperature rise and then substituting them into the equation derived by Lee and Cahill [76] will get the thermal conductivity of thin films.

1.5.5 Inverse Heat Conduction Problems

Over the past three decades, a considerable amount of work has been done on the study of inverse heat transfer problems by either analytical methods or numerical methods. The analytical methods include exact methods, polynomial methods, and integral methods [77]. These methods are only useful for solving linear one-dimensional problems with particular initial and boundary conditions. Numerical

inverse problems is that they are ill-posed [78]. By contrast, a well-posed problem meets the following three requirements: existence, uniqueness and stability. It has been proved that solutions to inverse heat conduction problems usually exist and are unique. However, the obtained estimates are not always numerically stable [77-80]. In other words, small inaccuracies in the measured interior temperatures may cause large oscillations in the calculated surface conditions. Thus, many special methods have been proposed to solve inverse heat conduction problems (IHCPs) [81-84]. The main purpose of these inverse methods is to improve the stability of numerical calculation results.

1.6 Objectives

This study primarily deals with microscale heat transfer in solids. First, it discusses transient heat transfer phenomena in solid dielectric thin films from both macroscopic and microscopic points of view. Phonon radiative transfer model as well as Fourier law and thermal wave theory are utilized to analyze the transient heat conduction phenomena in dielectric thin films. Diamond and GaAs thin films are chosen as the examples to demonstrate the differences between the macroscale and microscale heat transfer models.

Second, we examine microscale heat transfer in multilayer structures. The size

Ordinate Method [85] is adopted to solve the integro-differential equation. Besides the influences of size and curvature on thermal conductivity, the size and curvature effects on interface thermal resistance are also studied. Furthermore, the numerical predictions are compared to the experimental data to demonstrate the validity of the microscale heat conduction model.

Third, we examine microscale heat transfer in two-dimensional micro tubes. The past literature contains many theoretical and experimental studies of thermal characteristics of thin films, but very few of them paid attention to the thermal characteristic of micro tubes. There is no existing study, which shows that the size (thickness and height) and the curvature effects will change the effective thermal conductivity for micro tubes. In this study, the EPRT is employed to analyze the transient heat transfer in micro tubes. The influences of size and curvature on the thermal conductivity are examined closely.

By extending the foregoing analysis, we examine size effects on the performance of thermoelectric micro coolers. Superlattice structures are widely used to create high performance thermoelectric device because they can greatly reduce the phonon thermal conductivity of materials without much degrading the power factor (S²σ_e).

In this study, microscale heat transfer in thin film, nano wire and nano tube superlattices is simulated and then the effective thermal conductivity is calculated.

Understanding the influence of device sizes on the their thermal conductivity gives

systems. In this study, a space- marching technique [84] is adopted to estimate the temperature distributions and unknown boundary conditions from internal measurements. A radiation-boundary-condition model based on the AMM is employed to consider the interface thermal resistance between the thin- film and substrate. The influences of interface thermal resistance, measurement errors, and measurement locations are studied. Furthermore, the results of the estimation for the cases with or without interface thermal resistance are also compared.

Table 1.1 General features of heat carriers [3]

Free Electron Phonon Photon

Generation valence or excited electrons

lattice vibration

atomic, molecule transition

Propagating Media

in vacuum

or media in media in vacuum or media

Statistics Fermion Boson Boson Frequency 0~infinite Debye cut-off 0~infinite Dispersion ^{E =h2q2 (2m) E =E(q)} ν c= λ

Velocity (m/s) ~10⁶ ~10³ ~10⁸

Table 1.2 Characteristic time- and length scales and corresponding transport phenomena of energy carriers [38]

Length Scales Time Scales

Wavelength Mean Free Path

Relaxation Length

Diffusion Length Collision

Time

Wave Transport

Mean Free Time Relaxation

Time

Wave

Transport Microscopic Particle Transport Theory

Diffusion Time

Macroscopic Transport

Fig. 1.1 Schematic diagram of single-walled carbon nanotubes: (a) armchair, (b) zigzag, and (c) chiral [86].

(a) (b) (c)

Fig. 1.2 Schematic diagram of electric double layer [17].

Microchannel Wall

Ψ_s

Solution

Electric Do uble Layer

Fig. 1.3 Bright- field transmission electron micrograph of Cu nanoparticles dispersed in ethylene glycol [87].

(a)

(b)

(c)

T

H

Hot Side

I I

N P

Cold Side

I I

T

C

T

_C

Cold Side

I I

N P

Hot Side

I I

T

_H

(a)

(b)

N P N P N P N P

I I

Cold Side

Hot Side

ceramic plate thermoelements

conducting strip

Heater/Thermometer

Thin Film

I(ω)

V(3ω)

2. MICROSCALE HEAT TRANSFER IN SOLID THIN FILMS

Microscale heat transfer in solid thin films is attracting many researchers’ notice in recent years. To find out the difference between macroscopic and microscopic points of view is of great importance. In this chapter, diamond and GaAs thin films are chosen as the examples to illustrate the discrepancy between the macroscale and microscale heat transfer models. Fourier law, thermal wave theory and phonon radiative transfer model are utilized to analyze the transient heat conduction phenomena in solid thin films. Comparison of the temperature distributions predicted by Fourier law and EPRT has been done to seek the boundary between macroscale and microscale transfer regimes.

2.1 Analysis

2.1.1 Mathematical Fo rmulation

Consider a dielectric thin film of thickness L with temperature T initially. At ₀ time t =0, the temperature at x =0 is risen to T . Meanwhile, the temperature at _r

L

x = is still maintained at T . The schematic diagram is shown in Fig. 2.1. Phonon ₀ radiative transfer model as well as Fourier law and thermal wave theory are utilized to analyze heat conduction in the dielectric thin film. The dimensionless variables are

Fourier Law

The governing equation of the above problem according to Fourier law is

2

with initial condition 0

and boundary conditions 1

Thermal Wa ve Theory

The equation of thermal wave theory can be written as

2

with initial conditions 0

and boundary conditions 1

t coll

Here f is the distribution function of phonons. The collision term, t coll

complicated and makes the Boltzmann equation difficult to be solved. By using the relaxation-time approximation, the Boltzmann equation becomes

R

Where f is the equilibrium phonon distribution, which follows the Bose-Einstein ⁰ distribution. At 1993, Majumdar transformed the Boltzmann equation to an equation of phonon radiative transfer (EPRT) in the form [31]

v R the equilibrium intensity which can be transformed by the following relation

∫

−

Once the intensity of phonons is solved from the above equation, the temperature

) (ω

D is the density of states, and subscript p is the polarization index.

2.1.2 Numerical Method Fourier Law

The governing system is solved using the finite difference technique with implicit algorithm in time and central difference in space. Thus, the difference equation becomes

2

Thermal Wave Theory

By using the technique of Laplace transform, the solution can be obtained



Let u =γ i+ ς and substitute it into the above equation. Eq. (2-17) is reduced to a

Let u =γ i+ ς and substitute it into the above equation. Eq. (2-17) is reduced to a

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