Open access peer-reviewed chapter

# Thermoelectricity from Macro to Nanoscale: Wave Behaviour and Non-Local Effects

By Aldo Figueroa Lara, Iván Rivera Islas, Víctor Hernández García, Jaziel Rojas Guadarrama and Federico Vázquez Hurtado

Submitted: October 24th 2017Reviewed: February 27th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.75998

## Abstract

In this chapter, the physical principles to be taken into account in thermoelectricity at the nanometre scale are discussed. We argue that the numerical methods must also be adapted to the emergence of new physical behaviours at that scale, namely, wave propagation of heat, diffusive-ballistic transition, nonlocal effects, among others. It is first shown that thermoelectric phenomena at the nanoscale can be described by introducing thermodynamic inertia and nonlocal effects. The transport equations are obtained from the thermodynamics of irreversible processes. After this, we introduce the Spectral Chebyshev Collocation method as a well-suited numerical method to deal with the new physical behaviours appearing at the nanoscale. We then show the use of these formalisms to analyse specific and interesting aspects of the optimization of pulsed thermoelectricity and coupled thermoelectric modules.

### Keywords

• electrothermal
• pulsed thermoelectricity
• nanometre length scale
• heat wave propagation
• size effects
• thermal optimization
• coupled thermoelectric modules

## 1. Introduction: irreversible thermodynamics of thermoelectricity

Heat, electron and hole transport and generation-recombination of electron-hole pairs are processes which determine the functioning of thermoelectric devices. In this section, the principles of irreversible thermodynamics of modelling of thermoelectric phenomena are exposed. The contributions to the entropy production in the stationary state due to the dissipative effects associated with electron and hole transport, generation-recombination of electron–hole pairs as well as heat transport are analysed.

The operation of thermoelectric devices occurs on the basis of the so-called cross effects. On the one hand, the Seebeck effect which arises when an external temperature difference is applied leading to a charge flux and, on the other hand, the Peltier effect which causes thermal fluxes in the presence of an applied voltage [1, 2]. This is also connected with the Joule and Thomson effects arising with the electrical current together with the non-equilibrium electrons and holes and electron–hole recombination phenomena. The latter becomes a thermal source causing an internal energy heterogeneity described through the internal energy balance equation [3].

The importance of electron–hole recombination must be remarked since the optimization of the device must be achieved in the regime of bipolar transport [4]. In this context, the problem of heat dissipation and its removal concerns the parameters that characterize the work of the device. When the dimension of the device goes to the nanometric length scale, nonlocal and memory effects must be taken into account [5]. This is achieved through the introduction of thermodynamic inertia in the constitutive equations of the dissipative fluxes and local dependence of transport coefficients, respectively. In the following, the constitutive equations of dissipative fluxes will be derived from the very principles of irreversible thermodynamics [6]. Then nonlocal and inertial effects will be introduced.

Let us consider the system constituted by two species (electron and holes) and the lattice. Firstly, define the electron and hole densities by nrtand prt. The continuity equations for the electrons (electric charge q) and holes (electric charge +q) are given by

qntJn=qR,E1
qpt+Jp=qR,E2

where Jnand Jpare electric charge fluxes of electrons and holes, respectively, qis the elementary electric charge and Ris the balance of generation and recombination of electron–hole pairs processes. Now we write the balance equation of total internal energyuincluding the contribution of electrons un, holes upand the lattice uL:

ut+Ju=EJnJp+P,E3

with u=un+up+uLand Ju=Jnu+Jpu+JLuthe total internal energy flux. The total electric field (external plus self-consistent field) is represented by E. Finally, Pis a source term which can include light energy transference to the lattice and other processes. We use the expression for the Gibbs equation for each of the components of the system in order to find the balance equation of the total entropy density. We begin by writing the corresponding Gibbs equation for electrons, holes and the lattice. They are

Tndsn=dunΦndn,Tpdsp=dup+Φpdp,TLdsL=duL,E4

In these equations,Φxis the electrochemical potential of speciesx=n,p. Explicit expressions for the electrochemical potentials are the following.

Φn=νn,Φp=νp+E5

being φthe total electric potential (external plus self-consistent field). At this point, it is convenient to mention that the self-consistent electric field is given by Poisson’s equation:

εφs=qNA+ND++np,E6

where εis the permittivity constant and NA,ND+are the densities of ionized acceptors and donors, respectively. It is now assumed that the components of the system are in thermal equilibrium, that is, Tn=Tp=TLT. The sum of Eqs. (4) yields the balance equation for the total entropy density. One gets

sTt+JTs=JTs1T+1TJnEn,+1TJpEp,+RΦnΦpT+PT.E7

In obtaining Eq. (7) use has been made of Eqs. (13). The total entropy density sTin Eq. (7) is then given by.

sT=sn+sp+sL.E8

The total entropy flux JTshas been defined as

JTs=1TJq+q1ΦnJn+q1ΦpJp,E9

and the heat flux Jqbecomes:

Jq=Ju.E10

The field Ex,in Eq. (7) is a generalized electric field given by Ex,=q1ΦxE, with x=n,p.

The right-hand side of Eq. (7), excepting the term P/T, is the entropy generation term divided by the temperature, that is,

=JnEn,+JpEp,JTsT+RΦnΦp,E11

and it has the form JiXi, where Jiand Xiare generalized fluxes and thermodynamic forces, respectively. This allows us to identify the dissipative fluxes, and therefore, the origin of irreversibilities in the system. The generalized fluxes are JnJpJTsRand the corresponding forces En,Ep,TΦnΦp.

The second law of thermodynamics demands that σ>0.This condition is satisfied if the thermodynamic forces and fluxes are linearly related as follows:

JnJpJTsR=Lnn0Lns00LppLps0LsnLspLss0000LRREn,Ep,TΦnΦp,E12

where the coefficients Lxyare the so called Onsager coefficients. They are determined through phenomenological arguments and obey the reciprocity Onsager relations, namely, Lxy=Lyx, with x,y=n,p,s,R. Eq. (12) are the constitutive equations of the system which make complete the description offered by Eqs. (13, 6) together with the caloric equation uT=cVT, where cVis the specific heat at constant volume. The constitutive equation, Eq. (12), contain well-known phenomenological laws: Ohm, Fourier, Fick, and Peltier and Seebeck effects. In the following section, we expose some additional considerations to be taken into account when the dimensions of the thermoelectric systems are in the nanometric length scale.

## 2. Non-local and memory effects

In this section, we address the problem of heat transport in a thermoelectric nanoscaled layer when an electric current circulates through it. At that length scale, nonlocal and memory effects must be included. The former are due to size effects on the transport coefficients when going to the nanometric scale and the second one become from the thermodynamic inertia of the system. Let us consider the thermoelectric system shown in Figure 1, and let us pay attention to one of the semiconductor branches, the ntype for instance.

The analysis of heat transport in that element of the device of Figure 1 is based on the constitutive equations obtained in Section 1, Eqs. (12). The equations explicitly read [7, 8, 9]

τeffqt+q=KL+ΠSELσLTxΠσLVx,E13
τJJt+J=SELσLTxσLVx,E14

where for the sake of simplicity, we have denoted the heat and electric charge fluxes as qand J, respectively. Inertia and no locality were introduced through the time derivatives of the heat and electric charge fluxes and the dependence of the transport and thermoelectric coefficients on the width of the layer, respectively. The equations have been written in one spatial coordinate since if the length Lgoes to the nanometric scale, the dimensions of the branch nin the yand zdirections become much larger than L. The times τeffand τJare the relaxation times of the heat and electric charge fluxes, respectively. K, σand SEare the thermal conductivity, the electric conductivity and the Seebeck coefficient, respectively. In Eqs. (13) and (14), they are denoted to depend on the length L. Expressions for the transport coefficients as functions of Lare obtained within the higher order dissipative fluxes framework of extended irreversible thermodynamics [10, 11]. The expressions are the following

KL=K0L22π2lp21+4π2lpL21,E15
σL=σ0L22π2le21+4π2leL21,E16

SEL=4π2S0le23L22πle/Larctan2πle/L11,E17

with lpand lethe mean free path of heat and electric charge carriers, respectively. K0, σ0and S0are the bulk thermal conductivity, electric conductivity and Seebeck coefficient, respectively. A hyperbolic-type transport equation for temperature can be obtained by introducing Eqs. (13)(14) in Eq. (3) (with P=0) and using the caloric equation for the inner energy. The procedure may be followed in Appendix A of reference [5]. The resulting equation is

αeff2Tt2+Tt=ν2Tx2+βJ2+ςJtTx,E18

where the dimensionless coefficients are defined as follows:

αeff=τeffτ,ν=KLτρCvL2,β=τJ02ρCvσLTh,ς=J0τeffSELρCvL.E19

The characteristic time τ=π2L2ρCv4KLis the diffusion time and This a reference temperature. Equation (18) is here solved numerically for a thermoelectric thin film (in the branch n, see Figure 1) subjected to a Dirichlet boundary condition in the hot side of the thermoelectric device and a Robin type one in the cold side. We choose the space domain as 1x1, in such a way that boundary conditions can be written as:

Txx=1=γJT,T1=1,E20

where the dimensionless coefficient γis defined as γ=SELJ0L/2KL. As the initial condition, we state that the device is at room temperature, that is, Tx0=1. Eq. (18) shows step-solutions which are very challenging numerically speaking. Thus, it was solved by using the numerical code based on the Spectral Chebyshev Collocation method described in Section 3. In Figure 2a, it is shown the time evolution of the cold temperature towards the stationary state. Since Silicon is a basic material for short and long scale devices, our departing results come from considering doped Silicon as working material, whose properties have been published before in [12]: K0=149Wm−1 K−1, σ0=35.5×103Ω−1 m−1, SE=440×106V K−1 and aE=88×106m2 s−1, where aEis the thermal diffusivity. The mean free path of the heat carriers and their mean velocity are [10] lp=40nm and v=3K/ρCplp, respectively. The mean free path of electric charge carriers was assumed to be of the order of the lattice constant of Silicon le=0.5nm. As hot side temperature, we use Th=373.1 K [12]. We present the results without further discussion. The temperature starts to decrease once the electric current is applied. The transient to the stationary strongly depends on L. As it can be appreciated in Figure 2a, as the Knudsen numbers δ(defined as lp/L) increases the wave behaviour appears. On the contrary, for small δ, the steady state is reached quickly through a relaxation process without any oscillation. The minimum reached temperature is about 160Kwhen δ=4. The response of the system to a short electric pulse superimposed to the stationary state obtained with the optimal electric current was also studied. The squared electric pulse with magnitude 3.5is applied at time t=50. The wave behaviour of temperature produces subsequent supercooling transient process once the electric current pulse has been applied. The duration of it exceeds the duration of the supercooling obtained with the pulse. The minimum reached temperature is about 300 K when δ=4. In Figure 2a, it can also be seen an overheating after the application of the current pulse (which occurs at the dimensionless time 50). This effect is due to the thermal inertia, and it is more pronounced for small values of Lfor which the thermal inertia plays a more important role. Figure 2b shows the evolution to the stationary state by omitting in the calculations the effect of the size Lon the Seebeck coefficient given by Eq. (17). The oscillating decay for small values of Lremains but, interestingly, the minimum temperature reached is much higher than that reached when the effect of the size Lon the Seebeck coefficient is included by means of Eq. (17). Thus, the thermal efficiency of the cooling, measured by the temperature difference reached, is much greater due to the influence of the size on the Seebeck coefficient.

## 3. Computational methods

In this section, we present a couple of numerical methods in order to solve the hyperbolic transport equation (Eq. (18)) that models the heat transport in thermoelectric thin layers. Since Eq. (18) cannot be solved analytically, a numerical approach must be accomplished. Typically, low-order numerical methods such as finite differences (FD) are used. However, because of the high temporal gradients (due to the source term with crossed derivatives) for the small scales in the problem, that is, when the heat wave propagation or ballistictransport is present, the FD method fails in providing satisfactory results. As commented previously, this kind of equations is very challenging and we opt for a high-order numerical method in order to find its solution. High-order (or spectral) methods have previously been used to study heat transport based on the Maxwell-Cattaneo-Vernotte equation giving a hyperbolic transport equation in macroscopic systems [14, 15] and in some microscopic devices [16]. Thus, in this section, we discuss the numerical approximations of the one-dimensional hyperbolic heat transport equation for a low- and a high-order schemes.

### 3.1. Finite differences

The code for the finite differences scheme is a standard one. It considers a forward difference and a second-order central differences for the first and second derivatives in time, respectively. In turn, for the second-order spatial derivative, a second-order central differences was used as well. The grid points were uniformly spaced, the time integration was an explicit with constant time step.

### 3.2. Spectral Chebyshev collocation

The high-order numerical code is based on the Spectral Chebyshev Collocation (SCC) method. The method departs by establishing a partial sum of Chebyshev polynomials. The partial sum is then considered to represent the solution of a partial differential equation (PDE). The solution of the PDE equation is satisfied exactly at the Gauss-Lobatto collocation points

xi=cosN,i=1,,N1,E21

where Ndenotes the number of points or the size of the grid. Thus, the solution is in the domain of the standard Chebyshev polynomials, that is {x| −1 < x < 1}. The partial sum of Chebyshev polynomials was inserted in the spatial derivatives of Eq. (18) obtaining expanded derivative matrices. The obtained coefficient equation system was solved by the matrix-diagonalization method in the physical space directly. A further explanation of the SCC method is found in [17, 18]. In order to compare directly with FD scheme, a coordinate transformation to interval to 0 < x < 1 was done. The time marching scheme was the same as the FD method.

### 3.3. Comparison

Figure 3a presents the steady-state temperature as a function of the spatial x-coordinate in the thermoelectric film for the micro-scale. We can easily observe that the temperature difference between the hot (Th) and cold (Tc) sides is about 3.1 degrees. Because of the Joule effect,the temperature profile is parabolic. The cold side temperature Tcas a function of time is shown in Figure 4b. The departing point is the initial condition when Tc = 373.1 K, then the cooling Peltier effect acts till the temperature reaches the steady state Tc = 370 K. We can observe (in Figure 3) that both numerical schemes agree quantitatively in the steady state spatial distribution as well as modelling the transitory state. If we define the error between both solutions as

ε=maxuFDuSCC.E22

The maximum error for the steady profile (Figure 3a) with N = 30 is ε=8.6×105, and the error for the transitory is one order of magnitude higher, that is ε=2.5×104which is acceptable.

As the length of the system Ldiminishes, reaching the nanoscale, the steady state is a line with positive slope, see Figure 4a, whereas the wave heat transport is clearly visible during the transient, showing a damped harmonic oscillation, see Figure 4b. At this scale, the αeff coefficient in the heat Eq. (18) becomes important, and thus, the wave term becomes dominant, see Table 1. The overall error for the steady-state solution is acceptable (ε=9.8×105). However, it is considerable larger (ε=6×103) for the transient. Finally, we can conclude that the SCC technique is more robust since the convergence of the solution is assured with smaller grid points than the FD method.

L(m)αeffβγ
1 × 10−42.16 × 10−87.81 × 10−37.38 × 10−3
1 × 10−52.16 × 10−67.81 × 10−57.38 × 10−4
1 × 10−62.16 × 10−48.27 × 10−77.82 × 10−5
1 × 10−72.16 × 10−22.39 × 10−82.26 × 10−5
1 × 10−82.16 × 1002.00 × 10−91.89 × 10−5
1 × 10−92.16 × 1021.97 × 10−101.85 × 10−5

### Table 1.

Dimensionless coefficients in Eqs. (18) and (20) as a function of the length of the system.

## 4. Optimal performance of thin thermoelectric layers

The control of heat in the nanoscale could have important consequences in applications as refrigeration, energy generation, energy transport and others. So, nanophonics has become a very active field of theoretical, computational and experimental research in the last 15 years. Many questions about the use of non-equilibrium thermodynamics principles at the micro and the nanoscale are being discussed, and several issues should be solved to make devices at those scales a matter of practical use. Particularly, devices in the micro and nanometric length scales work at high frequencies and they generate heat fluxes that can be in the order of thousands of watts per square centimetre. These irreversible processes elevate temperature reducing the device’s life time. This makes necessary the study of those operating conditions producing less dissipation, which often correspond to the minimum entropy production. Here we describe the effects of the width on the time evolution of temperature in thin thermoelectric layers and, particularly, on the thermal figure of merit and the entropy generation. The analysis is based on hyperbolic-type Eq. (18) describing the time evolution of dissipative flows including size effects on the thermal and electric conductivities. The hyperbolic Eq. (18) was solved with a numerical code explained in Section 3. The transition from the diffusive heat transport to the wave propagation regime is controlled by the system’s size when going from the micro to the nanometric scale of lengths.

When applying an electrical current to the thermoelectric system in order to obtain the maximum gradient between boundaries, the temperature distribution along the system’s length is parabolic as it can be seen in Figure 5a. Such current is named the optimal electrical current. In turn, the entropy generation is a parabolic decaying function showing its higher value at the cold end of the device. It is important to note that these profiles are invariant to the device length when applying the optimal current. Figure 5b shows the dependence of the temperature difference between the boundaries of the thermoelectric (Eq. (23)), the thermal figure of merit (Eq. (24)), the entropy generation (Eq. (25)), and, as functions of the length’s device (inset):

ΔT=ThTc,E23
ZTh=2ThΔTThΔT2,E24
S.x=SJJ2T+SF1T2dTdx2,E25

where the coefficients are SJ=J02/σThand SF=K/l2. In Figure 5b, we can note that devices with L<106m increase their thermal figure of merit, from ZTh=0.017(for L<106m) to ZTh0.4for L<108m.

We find that the steady-state thermal figure of merit (TFM) improves as the width goes to the nanoscale while the entropy generation increases. We identify a transition at a length of the order of the mean free path of heat carriers; in silicon it is about 40 nm. When going from the micro to the nanoscale, this transition is featured by an abrupt increment of both the total entropy production and the TFM. Above L = 40 nm, the heat transfer is dominated by diffusive processes; below this value, it is in the form of heat waves. The wave heat transport is clearly visible in the nonstationary process. An interesting study of the ballistic-diffusive transition in metals controlled by the wave number vector can be seen in [20].

## 5. Pulsed thermoelectric phenomena in thin films

As it was shown in Section 2, pulsed regimes produce a lower temperature than that obtained in the stationary state even with the optimal electric current for both uniform and non-uniform materials. This phenomenon is due to the fact that the Peltier effect occurs mainly at the cold junction while Joule heating is distributed in the bulk introducing a difference in the time taken by each one to influence the cold side of the device. The cold temperature is first changed by the Peltier effect and after diffusion Joule heat reaches the cold junction affecting it. Some examples of devices which need to be overcooled during a short time are mid-IR laser gas sensors [21], condensation hygrometers and microelectronic processors generating hotspots [22, 23, 24, 25, 26]. The effect of the pulse form has been widely studied in macroscale of lengths. It has been shown that by applying a quadratic pulse form, the supercooling effect can be improved over other forms [27]. Some other pulse forms present additional advantages [28, 29]. Here we explore the influence of the electric pulse shape in the supercooling effect when the dimensions of the thermoelectric device go to the submicrometre length scale. We study the effects of the shape of the electric pulse on the maximum diminishing of temperature by applying pulses in the form tawith abeing a power going from 0 to 10.

In Figure 6, it can be seen the different shapes of the imposed electric pulse as a function of time. The duration in all cases is 0.163 and the maximum magnitude 3.5 over the stationary electric density (with normalized magnitude of one). The values are the optimal in order to obtain the maximum supercooling for the squared shape (t0). In the same figure, t0denotes the squared-shape pulse.

In Figure 7, it can be seen the time evolution of the temperature during the supercooling at the cold side Tcof the thermoelectric for two distinct thickness of the film, namely, (a) L=1×104m and (b) L=1×108m. Each curve corresponds to one of the shaped pulses accordingly with the notation of Figure 6. The curves in Figure 7a reproduces the previous result found in [23] (microscopic case). The curves in Figure 7b are the result of the present analysis. It is remarkable the fact that the super cooling effect is about eight times larger at nanometric than at micrometric scale.

Pulses with a fractionarynumber perform better for nanoscaleddevices, whereas those with abigger than unity do it for microscaledones. We also find that the supercooling effect is improved by a factor of 6.6 over long length scale devices in the best performances and that the elapsed supercooling time for the nanoscaleddevices equals the best of the microscaledones.

## 6. Coupled thermoelectric modules

Thermoelectric systems are efficient devices for small size cooling objectives. Thus, this section is devoted to study the thermal performance of system composed of two thermoelectric devices. As a first step, a theoretical model of the heat transport in both thermoelectric devices will be developed. As a second step, the theoretical results will be compared with experimental data. Similar experimental devices have been previously reported in the literature [31, 32, 33, 34, 35, 36]. However, we use a local approach instead of global energy balances, which allow us to obtain spatial distributions of the main physical properties.

### 6.1. Experimental procedure

The experimental set-up is conformed by a two-stage Peltier cooler, that is, two thermoelectric modules connected electrically independent and thermally in series, see Figure 8. The modules (with side length L = 30 mm and width h = 3.6 mm) are denoted by Min the figure and are made of Bismuth Telluride alloys. The two-stage system is located on a metal plate (15 × 8 cm). The plate is kept at a constant temperature by the contact with hot water, which is continuously forced to circulate by a pump in a rectangular frame where the hot plate is located. The pump is a LMI MILTON ROY Microprocessor dosing and is denoted by B. The modules are operated by an electric current Ifrom a BK PRECISION 1696 DC power supply V. The electric currents are in the range 0 < I < 1.517 A. Avoiding the transient, the temperature on top of module one M1was measured with a thermocouple (K type, Extech 470 True RMS Multimeter). Silitek thermal paste was used for the joints between the thermocouple and the cold wall, and the two modules. The latter is for avoiding the thermal decoupling of joints. When the two-stage system is turned on, the temperature in the cold wall is diminished because the modules generate a heat flow from the cold to hot wall. The error in the temperature data was obtained by adjusting a normal distribution to the data at same points.

### 6.2. Mathematical model

The heat transport problem can be reduced by considering only the heat flux in a branch (ntype) of the thin thermoelectric modules, see Figure 9. An electric current Ii, i = 1, 2, is injected through each of the thermoelectric modules. The system is thus subjected to a Dirichlet boundary condition in the hot side at Th, and a Robin type one in the cold side Tc, see Eq. (20).

At this point we assume that each of the thermoelectric modules behaves independent, as the single thermoelectric case (Eq. (18)), and they only share a common boundary. The heat transport for the coupled system is modelled by a system of two one-dimensional differential equations, each for every thermoelectric module

0=ν1d2T1dx2+β1J12,0=ν2d2T2dx2+β2J22.E26

The subscript denotes the first (1) and second (2) modules. The boundary conditions for the cold and hot sides are, respectively,

dT1dxx=1=γ1J1T11,T21=1.E27

While the coupling boundary conditions are

γ1J1T10dT1dxx=0=δLδKγ2J2T20δLδKdT2dxx=0,T10=T20,E28

being δL=L1/L2and δK=K2/K1the ratios between the lengths and the thermal conductivities of modules 1 and 2, respectively. The coupling boundaries at x = 0are obtained from a heat balance and by equalling the temperatures of both devices. The solution to the system of Eqs. (26) reads

T1x=a1x2+b1x+c1,T2x=a2x2+b2x+c2,E29

where the coefficients ai, biand ciare given by.

ai=βiJi22νi,c1=δLδKa21+a12+γ1J11+γ1J1γ1J121+γ1J1δLδK1+γ2J2=c2,b2=1a2c2,b1=c1γ1J1δLδKγ2J2+δLδKb2.E30

The difference of temperature between the hot and cold side is

ΔTcoupled=T21T11.E31

### 6.3. Comparison

Before presenting and discussing the theoretical results, the experimental measurements for a single and a coupled system are shown in Figure 10a, b. Figure 10a shows the temperature difference between the hot and cold sides ΔTas a function of the circulating electric current for a single thermoelectric device. A parabolic behaviour with a maximum ΔT25 K can be appreciated. Figure 10b presents elliptic isocurves of ΔTas function of the two electric currents, in the case that the system is composed of two thermoelectric modules. For this array, the maximum of the temperature difference is ΔT34 K, that is, nine degrees of extra cooling are obtained when using two thermoelectric modules instead of a single one. For both cases, the single and two thermoelectric module system, the electric current is normalized with I0=0.93 A, which is the optimal current for a single module.

Figure 10c shows the temperature difference ΔTbetween the hot and cold sides as a function of the electric current for a single thermoelectric device. This result can be compared with the experiment, Figure 10a, and the parabolic behaviour with a maximum is found. Figure 10d makes evident the dependence of ΔTwhen the system is composed on two thermoelectric devices, presenting elliptic isocurves. Theoretical results (Figure 10c, d) obtained from the simple model discussed before agree qualitatively with the experimental ones (Figure 10a, b) showing a maximum ΔTfor certain values of the electric currents. Since the theoretical model used to get Figure 10c, d is a one-dimensional representation of a three-dimensional problem, a quantitatively comparison is far from reflecting the experimental measurements. However, if we calculate the percentage of the maximum extra cooling obtained when using two thermoelectric modules instead of a single one, that is ΔTcoupledΔTsingle/ΔTsingle, we find that it is the same for both the experimental and the theoretical results. The maximum extra cooling that could be reached is 36%.

Some conclusions derived from the above are the following. First, the theoretical solutions of the temperature difference between the hot and the cold sides show a good qualitative agreement with the experimental measurements. Second, an optimal cooling with respect to the electric currents circulating through the modules has been found. Finally, an improvement of 36% in the performance, measured as mentioned above, of the coupled thermoelectric modules with respect to a single thermoelectric module is theoretically predicted. The analysis made in this section can be useful for the design of thermoelectric coolers.

## 7. Discussion and conclusions

We have analysed different aspects of the performance of thermoelectric films when an electric current flows through them, maintaining one of their sides at a constant temperature. First, we considered the effects of the size of the film on the Seebeck coefficient and the thermal and electrical conductivities as well as thermal inertia. This forms the basis for all the analysis of the problem of reducing the dimensions to the nanoscale. The introduction of such effects was done through the use of Eqs. (15)(17) in the constitutive equations of the irreversible thermodynamics of electric charge and heat transport, Eqs. (12). When the latter are combined with the conservation equations of charge and energy, Eqs. (1)(3), we arrive at the heat transport equation that was used systematically throughout the development of the analysis, Eq. (18). The different effects studied, namely, the wave behaviour of the propagation of heat, the response of the material to a pulse of electric current, the coupling of thermoelectric couples, the transition from the diffusive regime of heat transport to wave propagation, and so on were explained in terms of Eq. (18). The conclusions were diverse: (1) thermodynamic inertia improves the thermal performance of a cooler, (2) the transition from diffusive transport to wave propagation of heat is controlled by the size of the thermoelectric material, (3) in the operating pulsed mode in a cooler, the shape of the electrical pulse is crucial for its thermal performance, (4) it is always possible to find values ​​of the thermal parameters of a device that optimize its performance, (5) the operation of thermoelectric systems is closely related to the production of entropy in the system. Behind all these behaviours is the combined effect of the reduction of the thermal and electrical conductivities of the material and the increase of the Seebeck coefficient with the reduction of the dimensions. To close this section we show in Figure 11 the behaviour of the thermoelectric figure of merit, ZT=σSE2T/K, when the thickness of a doped silicon semiconductor film decreases towards the nanoscale of lengths. As can be seen in the figure, the increase in the Seebeck coefficient dominates the reduction of thermal and electrical conductivities. The net effect is the increase of the thermoelectric figure of merit. Notably, the figure at a length of l=109mis greater than that at l=105mby a factor of 287.

## 8. Perspectives

Undoubtedly, a work perspective in thermoelectric power generation and refrigeration systems is to bring to the nanoscale the dimensions of the elements that make up structured systems. There is a lot of work done in systems composed or structured from elements in the millimetric escale of lengths or in larger scales [38], but a very promising perspective is the construction of those energy converting systems from nano-components such as for instance, the case of 1D phononic crystals. Several problems on thermoelectric phenomena that require attention when going from the macro to the nanoscale are: effects of size on the thermoelectric properties of the components of the nanostructured systems, the wave character of the propagation of heat and the phenomena of thermal interference and resonance, the effect of wave propagation of heat in pulsed systems, the effects of size on irreversible processes, the simultaneous consideration of the effects of size and the temperature dependence of the properties of materials, the effect of thermal inertia on the wave propagation of heat. It is also necessary to study the relationship between entropy generation and thermoelectric conversion efficiency and between structure and efficiency. In relation to this, a working hypothesis is that it is possible to find a structure that minimizes the production of entropy and to exploit interference and resonance phenomena to improve the thermoelectric energy conversion efficiency.

## Acknowledgments

F.V. and A.F. acknowledge financial support from CONACYT-México under contracts 183358 and 258623, respectively. F.V. also thanks the project “Fronteras de la Ciencia” 367. A. F. also thanks the Cátedras program from CONACYT.

## Conflict of interest

We declare that there is no conflict of interest.

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Aldo Figueroa Lara, Iván Rivera Islas, Víctor Hernández García, Jaziel Rojas Guadarrama and Federico Vázquez Hurtado (November 5th 2018). Thermoelectricity from Macro to Nanoscale: Wave Behaviour and Non-Local Effects, Bringing Thermoelectricity into Reality, Patricia Aranguren, IntechOpen, DOI: 10.5772/intechopen.75998. Available from:

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