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

## 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

where

with

In these equations,

being

where

In obtaining Eq. (7) use has been made of Eqs. (1–3). The total entropy density

The total entropy flux

and the heat flux

The field

The right-hand side of Eq. (7), excepting the term

and it has the form

The second law of thermodynamics demands that

where the coefficients

## 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

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]

where for the sake of simplicity, we have denoted the heat and electric charge fluxes as

with

where the dimensionless coefficients are defined as follows:

The characteristic time

where the dimensionless coefficient ^{−1} K^{−1}, ^{−1} m^{−1}, ^{−1} and ^{2} s^{−1}, where *L* remains but, interestingly, the minimum temperature reached is much higher than that reached when the effect of the size

## 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 *ballistic* transport 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

where *N* denotes 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 *Tc* as 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

The maximum error for the steady profile (Figure 3a) with *N* = 30 is

As the length of the system *L* diminishes, 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 *α*_{eff} 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 (

L(m) | |||
---|---|---|---|

1 × 10^{−4} | 2.16 × 10^{−8} | 7.81 × 10^{−3} | 7.38 × 10^{−3} |

1 × 10^{−5} | 2.16 × 10^{−6} | 7.81 × 10^{−5} | 7.38 × 10^{−4} |

1 × 10^{−6} | 2.16 × 10^{−4} | 8.27 × 10^{−7} | 7.82 × 10^{−5} |

1 × 10^{−7} | 2.16 × 10^{−2} | 2.39 × 10^{−8} | 2.26 × 10^{−5} |

1 × 10^{−8} | 2.16 × 10^{0} | 2.00 × 10^{−9} | 1.89 × 10^{−5} |

1 × 10^{−9} | 2.16 × 10^{2} | 1.97 × 10^{−10} | 1.85 × 10^{−5} |

## 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):

where the coefficients are

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 *a* being 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 (

In Figure 7, it can be seen the time evolution of the temperature during the supercooling at the cold side

Pulses with a *fractionary* number perform better for *nanoscaled* devices, whereas those with *a* bigger than unity do it for *microscaled* ones. 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 *nanoscaled* devices equals the best of the *microscaled* ones.

## 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 *M* in 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 *I* from 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

### 6.2. Mathematical model

The heat transport problem can be reduced by considering only the heat flux in a branch (*n* type) of the thin thermoelectric modules, see Figure 9. An electric current *I*_{i}, 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

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

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

While the coupling boundary conditions are

being *x = 0* are obtained from a heat balance and by equalling the temperatures of both devices. The solution to the system of Eqs. (26) reads

where the coefficients

The difference of temperature between the hot and cold side is

### 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

Figure 10c shows the temperature difference

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,

## 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.