# Thermal transport at the nanoscale

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## Thermoelectric effect

Thermoelectrics is defined as the science and technology associated with thermoelectric generation and refrigeration [4]. Thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa. The term “thermoelectric effect” encompasses three separately identified effects: Seebeck effect, Peltier effect and Thomson effect. All of them are described below.

Seebeck effect

An electric potential is generated within any isolated conducting material that is subjected to a temperature gradient; this is the absolute Seebeck effect [5].
Relative Seebeck effect is the electromotive force production and consequently an electric current in a loop of material consisting of at least two dissimilar conductors, when the two junctions are maintained at different temperatures. This is depicted in Figure 1.3.
At the atomic scale, an applied temperature gradient at the two ends of a thermoelectric material causes carriers to diffuse from the hot to the cold side. Mobile charge carriers transport to the cold side, while oppositely charged carriers and immobile nuclei remain at the hot side, giving rise to a thermoelectric voltage. This voltage urges charge carriers to move back to the hot side. When these two transport mechanisms come to equilibrium, steady state condition in the material is reached. Thus, if we have two different thermoelectric materials in contact, voltage is generated due to temperature difference, the so called electromotive (emf) force. In the case of closed circuit (see Figure 1.3), current flows through the conductors, if their junctions are maintained at different temperatures.
We define the Seebeck coefficient (S), which is a measure of the magnitude of an induced thermoelectric voltage in response to an applied temperature difference at the two ends of a thermoelectric material. In the case of two materials in contact, the relative Seebeck coefficient is defined as SAB=SB-SA. More details concerning Seebeck Coefficient can be found in the corresponding section.
In general, the Seebeck effect is described by the following equation: ∇V=−S∙∇T (1.1) where ∇𝑉 is the voltage gradient (V) between two points of a material with temperature gradient ∇𝑇 (K) and S is the Seebeck coefficient (V/K).

Peltier effect

The Peltier effect is the reverse phenomenon of Seebeck effect and it is assumed to be the origin of the foundation of thermoelectric cooling. Opposed to Seebeck, relative Peltier effect is the reversible change in the heat content at an interface between dissimilar conductors that results from the flow of current across it [5]. The rate of heat absorption (or liberation) at the junction (Q̇ ) between two conductors A and B is given by the following equation: Q̇=ΠAB∙I (1.2) where ΠAB is the relative Peltier coefficient (ΠAB=ΠA−ΠB, where ΠA,ΠB are the absolute Peltier coefficients of conductor A and B respectively (V)) and 𝐼 the applied electric current (A) (from A to B, as shown schematically in Figure 1.4). The Peltier coefficient is a measure of the amount of heat carried per unit charge and its sign depends on which junction is heated and which is cooled.
Figure 1.4: Peltier effect when current is applied between two different conductors. One junction is heated up and the other is cooled down by the same amount of heat Q (𝜫𝑨𝑩>𝟎).
As it can be deduced from the previous analysis, if a simple thermoelectric circuit is closed and a temperature gradient exists, then a current will flow in the circuit, due to Seebeck effect. This current, however, in its turn, will transfer heat from one side to the other, due to Peltier effect. Thus, there is a close relationship between these two effects, which is represented with the following equation: ΠΑΒ=Τ∙SAB (1.3)
Both the magnitude and the direction of the heat flow in a thermoelectric circuit do not depend on the junction (neither on its dimensions nor on the contact potential) between the two materials, but only on the type of the materials used. This is applicable both in Seebeck and in Peltier effects. Each material, depending only on its thermoelectric properties, contributes with a specific way in the resulting heat flow in the circuit.
It is worth noting that Peltier effect should not be correlated to Joule heating. Joule heating does depend on dimensions, it has a quadratic behavior with current and it is an irreversible process, contrary to Peltier effect which is thermodynamically reversible [5].

Thomson effect

The Thomson effect is the reversible change of heat content within any single homogeneous conductor in a temperature gradient, when an electric current passes through it [5]. It, practically, describes the heating or cooling of a current-carrying conductor with a temperature gradient, as it is schematically shown in Figure 1.5. The measured rate of Thomson heat conduction is proportional to the current intensity and to the temperature difference, according to the equation: Q̇=−μ∙I∙∇Τ (1.4) where μ is the Thomson coefficient (V/K), I is the current intensity (A) and ∇Τ is the resulting temperature gradient (K). Thomson effect can be either positive or negative. If heat is evolved, when current flows from the hotter to the colder side, a positive Thomson effect occurs (Figure 1.5). Oppositely, for the same conditions heat is absorbed and a negative Thomson effect takes place. Figure 1.5: Positive Thomson effect in a current-carrying conductor with a temperature difference (T1< T2).

Thermoelectric figure of merit

The thermoelectric performance of a material is characterized by its dimensionless figure of merit: zT=σS2T(ke+kl)⁄, (1.5) where σ is the electrical conductivity (S/m), S the Seebeck Coefficient(V/K), T the absolute temperature (K), ke and kl the electronic and lattice components of thermal conductivity (W/m.K) respectively. The factor σS2 is the well-known power factor. The need of improving the overall performance of thermoelectric materials was pointed out in the introduction of this chapter. It is highly demanding to find thermoelectric materials with high zT values. In this respect, high Seebeck coefficient and high electrical conductivity are needed while thermal conductivity should be low. Each one of these parameters is described below along with the most frequently used techniques for their measurement.

Seebeck Coefficient

Absolute Seebeck Coefficient (S), also known as thermopower or thermoelectric power, is a measure of the magnitude of an induced thermoelectric voltage in response to an applied temperature difference at the two ends of a thermoelectric material. It is a fundamental electronic transport property of a material and is defined as the ratio of thermoelectric voltage (ΔV) to the temperature difference (ΔΤ): S=−ΔVΔΤ (1.6)
Practically, Seebeck coefficient is a measure of the entropy that is transferred with a charge carrier as it moves in the volume of a thermoelectric material, divided by the charge itself [4]. Consequently, Seebeck coefficient is determined by the interaction of charge carriers with each other and with phonons.
From the above analysis it is deduced that the thermoelectric power S is the sum of a diffusion part Sd and a phonon drag part Sph, such that: S=Sd+Sph.
The diffusion part results from the spatial variation of the occupation probability of the charge carriers, which is caused by the temperature gradient along the sample. The phonon drag part is due to momentum that is transferred from lattice vibrations to electrons by electron-phonon scattering [6]. The separation of Seebeck coefficient in these two components is useful later for the interpretation of the measurements carried out on porous Silicon samples.

### Seebeck coefficient measurement methods

In Seebeck Coefficient metrology, the diversity in apparatus designs, data acquisition and contact geometry has resulted in conflict materials data [7]. Practically, it was just very recently when Lowhorn et al. from National Institute of Standards and Technology (NIST) reported the development of a certified Seebeck coefficient Standard Reference Material from 10 to 390K [8]. This is a bar-shaped piece of nonstoichiometric telluride.
In order to extract S of a thermoelectric material, a temperature difference must be applied and measured at two points of the sample under test with simultaneous measurement of the resulting thermovoltage. Then S is extracted straightforward from Equation (1.6). The basic requirements for obtaining an accurate Seebeck coefficient measurement are:
 Good thermal and electrical contacts
 Synchronous measurement of voltage and temperature
 Accurate measurement of small voltages and temperatures with eliminating external parasitic contributions
Extracting reliably the Seebeck coefficient of a material and, especially when it is in thin film form, is not an easy task. In the following section, the relevant measurement methods and possible arrangements are summarized [9, 10].
Although the dimensions of a sample do not affect its S, the accuracy of the measurement depends on the used configuration. There are two possible such arrangements; two and four-point probe.
 Two point-probe configuration
In two-point probe configuration, the temperature difference and the thermoelectric voltage are measured on the probes (hot and cold sinks), which are in direct contact with the ends of the sample, as it is shown in Figure 1.6. Such arrangement is preferred since it is simpler and avoids contacts problems [9].
 Four-point probe configuration
In four-point probe configuration, the temperature and thermoelectric potential are measured at two points of the sample and separately from the hot and cold sinks, as Figure 1.7 depicts. This arrangement is convenient when the same platform is used for the electrical conductivity measurements, because with the four-point probe this measurement is far more accurate, as it will be described in the relevant section.
Each configuration described above can be used in either integral or differential Seebeck coefficient measurement mode [9]. Both of them are reviewed below.
 Integral method
In the integral mode, large temperature differences are applied at the two ends of the sample under test and the corresponding thermovoltage is measured. An analytic approximation is then used to fit the experimental data. The Seebeck coefficient is obtained by differentiating these data. This method minimizes the influence of voltage offsets due to large temperature gradients; corresponding errors are thus minimized. However, due to the complex temperature dependence of S, there is no simple analytical expression to be used for fitting the experimental data. Moreover, there are not objective criteria to evaluate the accuracy of the obtained derivative. To overcome the drawbacks of this method, the differential one can be used instead.
 Differential method
In this method, a small temperature gradient is applied between the two ends of the thermoelectric material under test and the induced potential is simultaneously measured. Limiting the measurements in a short enough temperature range, the Seebeck coefficient is just the ratio of voltage difference to the temperature gradient (considering linear dependence of Seebeck coefficient with temperature at this specific range). Even though the differential method solves the problems related to the integral method, it demands much more sensitive instrumentation because the gradients measured could be very small.
The differential method is distinguished in three categories: steady-state, quasi-steady-state and transient. A brief description of each of these categories is given in the following paragraphs.
As its name indicates, measurements are recorded after the steady-state condition in the thermoelectric system is reached. Then, multiple measurements are taken and the Seebeck coefficient is extracted from the linear fit of thermovoltage vs temperature difference data points. This way, the possible voltage offsets are eliminated. Although the method applying steady-state conditions is proved to be accurate enough, it is time consuming and thus impractical and inefficient. An improved alternative is the use of quasi-steady state condition.
In this case, multiple measurements are recorded before a steady-state condition is reached. Heat flux increases continuously during data recording. It is essential that instrumentation and data acquisition are carefully designed and used because thermal offsets could be introduced. A common solution to diminish such effects is to use interpolated data for the time dependence of temperature and voltage [11].
 Transient condition
When the transient method is used, a sinusoidal temperature difference is applied between the two ends of the thermoelectric sample. Then lock-in amplifiers are used to extract the voltage and temperature difference and consequently Seebeck coefficient. While faster, this technique has some limitations concerning the thickness of the samples. In addition, there are no objective criteria on the appropriate frequency range for an accurate measurement. Moreover, the requirement of lock in amplifiers increases the cost and adds complexity to data acquisition.

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Thermal conductivity

Thermal conductivity k is the property of a material to conduct heat. Heat energy is transmitted through solids via electrical carriers, phonons, electromagnetic waves, spin waves or other excitations. The total thermal conductivity of a material is the sum of all the components representing the different excitations that occur in the solid [12]. However, the main heat carrying entities in solids are charge carriers and phonons, with other excitations contributing much less to the total thermal conductivity. Thus, in most cases other contributions are considered to be negligible. This assumption leads to the simplification that thermal conductivity of a solid is composed of two independent terms: the electronic (𝒌𝒆) and the lattice (𝒌𝒍) parts, such that: k=ke+kl (1.7)
According to Fourier’ s law of conduction, thermal conductivity is defined as the rate of heat flow through a square meter of a solid where a temperature gradient exists in the direction of heat flow and it is given by the following equation: k=−𝑞̇⃗∇⃗⃗⃗T (1.8) where 𝑞̇⃗ is the heat flux vector (W/m2) across a unit cross section perpendicular to 𝑞̇⃗ and T is the absolute temperature (K).
Thermal conductivity, and particularly its temperature dependence, is related to the structure and morphology of solids. Furthermore, thermal conductivity is a key parameter in the characterization of thermoelectric materials. Thus, its accurate measurement is necessary for the estimation of the overall thermoelectric performance of a material.

#### Thermal conductivity measurement methods

The accurate measurement of this parameter can pose many challenges. The structure of the sample, its dimensions and the thermal conductivity value itself should be taken into account in order to minimize measurement errors. Particularly, when sizes go down to nanometer scale and/or thermal conductivity values go to extremes, practical and other issues may be arise.
This section provides a review of the more typical measurement methods for thermal conductivity characterization, along with their advantages and drawbacks. Thermal conductivity measurement techniques can be categorized according to the diagram of Figure 1.8 and are comprehensively reviewed in Ref. [12]. The general principle of all the measurement techniques is heating of the sample (either electrically or optically) and then measuring heat transport through the material under test.
As it can be seen in Figure 1.8 there are electrical and optical heating and sensing methods with the former being direct and the latter indirect. Generally, using direct methods, the amount of heat transfer into the sample can be precisely controlled and the temperature increase due to the heat pulse can be accurately determined. Thus, electrical methods are more frequently used for such measurements.
The electrical direct methods can be distinguished in steady-state and transient methods. Both of them were used and compared in this thesis. In the following, a brief description of each method will be given.

Introduction
Chapter 1 Thermoelectric materials and devices
1.1 Introduction
1.2 Thermoelectric effect
1.2.1 Seebeck effect
1.2.2 Peltier effect
1.2.3 Thomson effect
1.3 Thermoelectric figure of merit
1.3.1 Seebeck Coefficient
1.3.1.1 Seebeck coefficient measurement methods
1.3.2 Thermal conductivity
1.3.2.1 Thermal conductivity measurement methods
1.3.3 Electrical conductivity
1.3.3.1 Electrical conductivity measurement methods
1.4 Optimization of the thermoelectric performance of a material
1.4.1 State-of-the-art thermoelectric materials
1.5 Si-based thermoelectrics
1.5.1 State-of-the-art Si-based thermoelectric materials
1.6 Thermoelectric applications
1.6.1 Thermoelectric generator
1.6.2 Thermoelectric cooling
1.6.3 Sensing
1.7 Conclusions
Chapter 2 Formation and Properties of Porous Silicon
2.1 Introduction
2.2 Porous Si formation
2.2.1 Porous Si formation by electrochemical etching of bulk c-Si
2.2.1.1 Fundamentals of electrochemical etching of c-Si
2.2.1.2 I-V Curves and formation conditions
2.2.1.3 Anodization cells
2.2.2 Porous Silicon formation by electroless etching
2.2.2.1 Stain etching
2.2.2.2 Galvanic etching
2.2.2.3 Metal-Assisted Chemical Etching (MACE)
2.2.3 Local porous Si formation
2.2.3.2 Local anodization using etch stops
2.2.4 Free standing porous Si membranes (FSPSi)
2.3 Porous Si morphology and structure
2.4 Porosity and thickness measurements
2.5 Conclusions
Chapter 3 Porous Si thermal conductivity in the tPorous Si thermal conductivity in the temperature range 4.2emperature range 4.2–350K350K
3.1 Introduction
3.2 Thermal transport at the nanoscale
3.3 Temperature dependence of thermal conductivity
3.3.1 Crystalline materials
3.3.2 Amorphous materials and glasses
3.4 Porous Si layers studied in this thesis
3.4.1 Isotropic Porous Si
3.4.2 Anisotropic Porous Si
3.4.3 Test structure
3.5 DC method with consequent FEM analysis
3.5.1 Finite Element Method (FEM) simulations – General considerations
3.5.1.1 Heat transfer
3.5.2 Implementation of the method
3.5.3 Experimental results – Temperature dependence of porous Si Thermal resistance
3.5.4 Experimental results – Temperature dependence of Porous Si thermal conductivity
3.5.5 Comparison with other materials and theory
3.6 3ω method
3.6.1 General assumptions and considerations
3.6.1.1 One – dimensional Line heater
3.6.1.2 Infinite heat source at the surface
3.6.1.3 Effect of finite thickness of the substrate
3.6.1.4 Finite width of the heater
3.6.1.5 Approximate solution to the exact equation
3.6.2 Experimental setup
3.6.3 Data analysis
3.6.3.1 The slope method
3.6.3.2 Extracting thermal conductivity using Cahill’s integral form – Approach1
3.6.3.3 3ω method with consequent FEM analysis – Approach 2
3.6.4 Experimental results
3.6.4.1 Bulk c-Si covered with a thin TEOS oxide
3.6.4.2 Anisotropic porous Si – 70% porosity
3.6.4.3 Isotropic porous Si – 63% porosity
3.7 Conclusions
Chapter 4 Thermal conductivity of PSi in the temperature range 4.2Thermal conductivity of PSi in the temperature range 4.2–20K 20K –– Interpretation of the plateauInterpretation of the platea
4.1 Introduction
4.2 Fractals and their physical properties
4.2.1 Fractal dimension
4.2.2 Methods of measuring fractal dimension
4.2.2.1 Scattering experiment
4.2.2.2 Image analysis
4.3 Fractons
4.4 Fractal nature of porous Si
4.4.1 Porous Si fractal dimension
4.4.1.1 Isotropic Porous Si
4.4.1.2 Anisotropic porous Si
4.5 Plateau-like behavior of porous Si thermal conductivity at cryogenic temperatures – Interpretation based on its fractal geometry
4.6 Conclusions
Chapter 5 Effectiveness of porous Si as a thermal insulating platform on the Si wafer Effectiveness of porous Si as a thermal insulating platform on the Si wafer
5.1 Introduction
5.2 Temperature distribution in a test device incorporating a thermal insulating porous Si layer
5.3 Effect of applied power
5.4 Effect of porous Si layer thickness
5.5 Comparison to bulk c-Si and other CMOS compatible thermal insulators
5.6 Conclusions
Chapter 6 Seebeck coefficient of porous Si as a function of porositySeebeck coefficient of porous Si as a function of porosity
6.1 Introduction
6.2 Study of porous Si free standing membranes
6.2.1 Fabrication process
6.2.2 SEM and TEM characterization
6.2.3 PL measurements
6.3 Seebeck coefficient measurements
6.3.1 Home-built setup description
6.3.2 Data analysis
6.3.3 Validity of the measurements
6.3.3.1 Seebeck coefficient of highly doped p-type Si
6.3.3.2 Diagnostic test measurements for hysteretic behavior
6.3.3.3 Comparison with simulations
6.4 Experimental results – Porosity dependence of porous Si Seebeck coefficient
6.5 Conclusions
Chapter 7 Thermoelectric performance of LPCVD polycryThermoelectric performance of LPCVD polycrystalline Si thin filmsstalline Si thin films
7.1 Introduction
7.2 Polycrystalline Silicon
7.2.1 Low Pressure Chemical Vapor Deposition (LPCVD)
7.2.2 Microstructure of undoped polysilicon films deposited by LPCVD
7.2.3 Microstructure of doped polysilicon films deposited by LPCVD
7.2.4 Polysilicon grain growth
7.3 Polycrystalline Si thin films studied in this thesis
7.3.1 Structural characterization
7.4 Electrical resistivity and TCR
7.4.1 Test structure
7.4.2 Measurement method
7.4.3 Experimental results
7.4.3.1 Electrical resistivity
7.4.3.2 TCR
7.4.3.3 Comparison with theory
7.5 Seebeck coefficient
7.5.1 Test structure
7.5.2 Measurement method
7.5.3 Experimental results
7.5.3.1 Comparison with theory
7.6 Thermal conductivity
7.6.1 Test structure
7.6.2 Measurement method
7.6.3 Experimental results
7.6.4 Comparison with theory
7.7 Power factor & Figure of merit
7.8 Comparison with other works on polysilicon films found in the literature
7.9 Conclusions
Conclusions
Perspectives

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