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.