**Kinetic Energy Harvesting**

Helios Vocca and Francesco Cottone

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/57091

## **1. Introduction**

[4] J.P. Sethna, Statistical Mechanics: Entropy, Order Parameters and Complexity, 6.4,

[5] Umberto Marini Bettolo Marconi; Andrea Puglisi; Lamberto Rondoni; Angelo Vul‐ piani (2008). "Fluctuation-Dissipation: Response Theory in Statistical Physics". Phys‐

[6] Macroscopic equations for the adiabatic piston, Massimo Cencini, Luigi Palatella, Si‐ mone Pigolotti, Angelo Vulpiani. Physical Review E (Statistical, Nonlinear, and Soft

Oxford Univ. Press. 2008

ics Reports 461 (4–6): 111–195.

Matter Physics), Vol. 76, No. 5. (2007).

24 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

The recovery of wasted energy present in the ambient that is a reject of artificial or natural processes to power wireless electronics is paving the way for enabling a huge number of applications. One of the main targeted technologies that meets the levels of harvestable power, typically few hundreds of microwatts, is represented by wireless sensor networks (WSNs) [1]. This technology consists of a grid of spatially-distributed wireless nodes that sense and communicate information like acceleration, temperature, pressure, toxicity of the air, biolog‐ ical parameters, magnetic field, light intensity and so on, among each other and up to the end user through a fixed server. In the next years, WSNs will be massively employed in a wide range of applications such as structural monitoring, industrial sensing, remote healthcare, military equipment, surveillance, logistic tracking and automotive monitoring. In fact, harvesting energy directly from the ambient not only represents a realistic mean to integrate or substitute batteries, but is the sole way for enabling many contemporary and future wireless applications that will be all integrated in the so called "internet of things" [2].

Actually, WSNs already have the characteristics of ubiquity, self-organizing and self-healing but they would not be deployable unless they will also be self-powering. As a matter of fact, it is very expensive and impractical to change batteries in most of the anticipated potential applications. For long-term operation in inaccessible or harsh locations, energy harvesting is a key solution. For example, long-term environmental, structural health of buildings or bridge monitoring and control would require many thousands of integrated sensors impossible to be replaced or maintained. The possibility for chronically ill patients to be continuously moni‐ tored without changing batteries would represent a significant improvement in their life quality.

Among various renewable energy present in the environment such as solar, radio frequency RF, temperature difference and biochemical, kinetic energy in the form of mechanical vibra‐

© 2014 Vocca and Cottone; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

tions is deemed to be the most attractive, in the low-power electronic domain, for its power density, versatility and abundance [3]. This type of energy source is located in buildings, vibrating machineries, transportations, ocean waves and human beings, and it can be con‐ verted to power mobile devices.

The power consumption of wireless sensors has been largely reduced in the last years thanks to the Ultra-Low-Power electronics [4]. Typical power needs of mobile devices can range from few microwatts for wristwatches, RFID, MEMS sensors and actuators up to hundreds of milliwatts for MP3, mobile phone and GPS applications. They are usually in a sleep state for the 99.9% of their operation time, waking up for a few milliseconds only to communicate data. Consequently, their average power consumption has been reduced below 10µW in order to match the power density capability of current generators (100-300 microwatts per cubic centimeter). For comparison, a lithium battery can provide 30µW/cc for 1 year or 30mW/cc for just 10 hours, while a vibration-driven generator could last for at least 50 years with the same power level [5]. Along with virtually infinite operational life, many other benefits come from motion-driven energy harvesting: no chemical disposal, zero wiring cost, maintenance-free, no charging points, capability for deployment in dangerous and inaccessible sites, low cost of retrofitting, inherent safety and high reliability.

A typical integrated vibration-powered wireless sensor includes an embedded vibration energy harvester (VEH), multiple-sensor module, microcontroller and a transceiver (Figure 1). Due to the variable nature of vibrations in their intensity and frequency, the device also contains an AC/DC voltage regulation circuit, which in turn can recharge a temporary storage system, typically a super-capacitor or a thin film Lithium battery. Capacitors are usually preferred as temporary storage systems for their longer lifetime, higher power density and fast recharging. In some applications, however, a storage system is not even necessary. The vibration energy harvester module is often tailored for the specific application and vibration spectrum of the source: harmonic excitation, random noise or pulsed movement.

## **2. Main conversion techniques**

There are three main categories of kinetic-to-electrical energy conversion systems: piezoelec‐ tric, electrostatic and electromagnetic. In addition, there is the magnetostrictive branch as a variant of piezoelectric except for the use of magnetically polarized materials [6]. Each technique presents advantages and drawbacks. Therefore, there not exist a technique suitable for all cases and the optimal choice depends on the specific application.

Piezoelectric transducers make use of electrically polarized materials such as Barium Titanate (BaTiO3), Zinc Oxide (ZnO) and Lead Zirconate Titanate (Pb[ZrxTi1−x]O3), commonly known as PZT which is considered one of the best materials for high electromechanical coupling. The direct piezoelectric effect used for energy harvesting was early discovered by French physicists Jacques and Pierre Curie in 1880. It occurs when an electric charge is generated within a material in response to applied mechanical stress (Figure 2). The strain and coupling coeffi‐ cients in the fundamental piezoelectric equations are in general much higher in 33 mode than

3 Figure 1. Wireless sensor network and vibration-driven wireless node with power fluxes. **Figure 1.** Wireless sensor network and vibration-driven wireless node with power fluxes.

9 suitable for all cases and the optimal choice depends on the specific application.

Running Title

tions is deemed to be the most attractive, in the low-power electronic domain, for its power density, versatility and abundance [3]. This type of energy source is located in buildings, vibrating machineries, transportations, ocean waves and human beings, and it can be con‐

26 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

The power consumption of wireless sensors has been largely reduced in the last years thanks to the Ultra-Low-Power electronics [4]. Typical power needs of mobile devices can range from few microwatts for wristwatches, RFID, MEMS sensors and actuators up to hundreds of milliwatts for MP3, mobile phone and GPS applications. They are usually in a sleep state for the 99.9% of their operation time, waking up for a few milliseconds only to communicate data. Consequently, their average power consumption has been reduced below 10µW in order to match the power density capability of current generators (100-300 microwatts per cubic centimeter). For comparison, a lithium battery can provide 30µW/cc for 1 year or 30mW/cc for just 10 hours, while a vibration-driven generator could last for at least 50 years with the same power level [5]. Along with virtually infinite operational life, many other benefits come from motion-driven energy harvesting: no chemical disposal, zero wiring cost, maintenance-free, no charging points, capability for deployment in dangerous and inaccessible sites, low cost of

A typical integrated vibration-powered wireless sensor includes an embedded vibration energy harvester (VEH), multiple-sensor module, microcontroller and a transceiver (Figure 1). Due to the variable nature of vibrations in their intensity and frequency, the device also contains an AC/DC voltage regulation circuit, which in turn can recharge a temporary storage system, typically a super-capacitor or a thin film Lithium battery. Capacitors are usually preferred as temporary storage systems for their longer lifetime, higher power density and fast recharging. In some applications, however, a storage system is not even necessary. The vibration energy harvester module is often tailored for the specific application and vibration

There are three main categories of kinetic-to-electrical energy conversion systems: piezoelec‐ tric, electrostatic and electromagnetic. In addition, there is the magnetostrictive branch as a variant of piezoelectric except for the use of magnetically polarized materials [6]. Each technique presents advantages and drawbacks. Therefore, there not exist a technique suitable

Piezoelectric transducers make use of electrically polarized materials such as Barium Titanate (BaTiO3), Zinc Oxide (ZnO) and Lead Zirconate Titanate (Pb[ZrxTi1−x]O3), commonly known as PZT which is considered one of the best materials for high electromechanical coupling. The direct piezoelectric effect used for energy harvesting was early discovered by French physicists Jacques and Pierre Curie in 1880. It occurs when an electric charge is generated within a material in response to applied mechanical stress (Figure 2). The strain and coupling coeffi‐ cients in the fundamental piezoelectric equations are in general much higher in 33 mode than

spectrum of the source: harmonic excitation, random noise or pulsed movement.

for all cases and the optimal choice depends on the specific application.

verted to power mobile devices.

retrofitting, inherent safety and high reliability.

**2. Main conversion techniques**

in 31 [7]. However the 33 mode of bulk crystal corresponds to very high natural frequencies (~1 to 100 kHz), while longitudinal strain is easily produced within a cantilever beam that resonates at lower frequencies (~100 Hz) (Figure 2c). 4 **2. Main conversion techniques**  5 There are three main categories of kinetic-to-electrical energy conversion systems: 6 piezoelectric, electrostatic and electromagnetic. In addition, there is the magnetostrictive 7 branch as a variant of piezoelectric except for the use of magnetically polarized materials [6]. 8 Each technique presents advantages and drawbacks. Therefore, there not exist a technique

4 ICT-Energy - Nanoscale Energy Management

2 Figure 2. (a) Direct piezoelectric effect with 33 and 31 strain-charge coupling. (b) 3 Polarization process scheme. (c) Drawing of bimorph piezoelectric cantilever beam. 4 Piezoelectric systems are capable of high voltage level (from 2 to several volts), well **Figure 2.** (a) Direct piezoelectric effect with 33 and 31 strain-charge coupling. (b) Polarization process scheme. (c) Drawing of bimorph piezoelectric cantilever beam.

5 adapted for compact size and very good in terms power density per unit of volume. 6 However, piezoelectric coupling decreases very fast at micrometric scale and relatively large 7 load impedances are required to reach the optimal working point [8]. Besides, other 8 problems must be considered such as aging, depolarization and brittleness. For low 9 frequency applications, like those related to wearable sensors, polymer-based materials (e.g. 10 dielectric elastomers) constitute a valid alternative to ceramics because of their flexibility,

12 Electromagnetic technique is simply realized, according to Faraday's law, by coupling a 13 static magnetic field produced by a permanent magnet and a solenoid in relative motion – 14 one of which usually acts as a stator; the other as a mover. These systems show 15 complementary behaviour in terms of frequency bandwidth and optimal load in relation to 16 piezoelectric techniques. They are recommended for lower frequencies (2-20 Hz), small 17 impedance and medium size [10]. Furthermore, their cost is smaller than other solutions. 18 Most of the commercial solutions are available at centimetre scales because they exhibit 19 higher power density than piezoelectric devices. On the other hand, the integration of 20 electromagnetic harvesters into micro-electro-mechanical-systems (MEMS) results difficult.

21 However, some of these limitations have been overcome to date [11-13].

1

1 2

11 inexpensiveness and durability [9].

Running Title

1

22

Piezoelectric systems are capable of high voltage level (from 2 to several volts), well adapted for compact size and very good in terms power density per unit of volume. However, piezoelectric coupling decreases very fast at micrometric scale and relatively large load impedances are required to reach the optimal working point [8]. Besides, other problems must be considered such as aging, depolarization and brittleness. For low frequency applications, like those related to wearable sensors, polymer-based materials (e.g. dielectric elastomers) constitute a valid alternative to ceramics because of their flexibility, inexpensiveness and durability [9].

Electromagnetic technique is simply realized, according to Faraday's law, by coupling a static magnetic field produced by a permanent magnet and a solenoid in relative motion – one of which usually acts as a stator; the other as a mover. These systems show complementary behaviour in terms of frequency bandwidth and optimal load in relation to piezoelectric techniques. They are recommended for lower frequencies (2-20 Hz), small impedance and medium size [10]. Furthermore, their cost is smaller than other solutions. Most of the com‐ mercial solutions are available at centimetre scales because they exhibit higher power density than piezoelectric devices. On the other hand, the integration of electromagnetic harvesters into micro-electro-mechanical-systems (MEMS) results difficult. However, some of these limitations have been overcome to date [11-13]. 5

2 Figure 3. (a) Simple architecture of em-VEH where a moving magnet (mover) oscillates with 3 respect to a fixed coil (stator). (b) Moving magnet across coil arrangement with discrete 4 components [11-13]. (c) Microfabricated em-VEH where a small magnet oscillates towards a 5 planar coil [11]. **Figure 3.** (a) Simple architecture of em-VEH where a moving magnet (mover) oscillates with respect to a fixed coil (stator). (b) Moving magnet across coil arrangement with discrete components [11-13]. (c) Microfabricated em-VEH where a small magnet oscillates towards a planar coil [11].

6 Electrostatic harvesters basically consist of a variable capacitor in which one electrode is 7 attached to an oscillating mass suspended by beams and the counter electrode is fixed 8 elsewhere in the support structure. When a force is applied to the mass either the dielectric 9 gap or the overlap surface of electrodes varies depending on the moving direction: the first 10 case being referred to as an in-plane gap-closing converter (Figure 4a), while the second case 11 as an overlap varying converter (Figure 4b). As a consequence, the capacitance changes and 12 additional charges occur at the electrodes in order to balance the bias voltage. Hence, during 13 the movement of the proof mass, a current flows through a load shunted to plates. A similar Electrostatic harvesters basically consist of a variable capacitor in which one electrode is attached to an oscillating mass suspended by beams and the counter electrode is fixed elsewhere in the support structure. When a force is applied to the mass either the dielectric gap or the overlap surface of electrodes varies depending on the moving direction: the first case being referred to as an in-plane gap-closing converter (Figure 4a), while the second case as an overlap varying converter (Figure 4b). As a consequence, the capacitance changes and

14 method of fixed bias voltage is that of charge constrained where a constant charge is held

16 One of the main disadvantages of electrostatic vibration harvesters is the need of an 17 external voltage source in order to be pre-charged. This fact seems to contrast with the goal 18 of energy recycling from the ambient, but makes sense if the source comes from the energy 19 storage associated to the harvester [12]. In that case the generator only needs to be kick-20 started at the beginning of the conversion process. Some designs overcome this problem by

15 into the plates by means of a battery or another capacitor.

21 using electrets to provide the pre-charge bias voltage [14].

additional charges occur at the electrodes in order to balance the bias voltage. Hence, during the movement of the proof mass, a current flows through a load shunted to plates. A similar method of fixed bias voltage is that of charge constrained where a constant charge is held into the plates by means of a battery or another capacitor.

Piezoelectric systems are capable of high voltage level (from 2 to several volts), well adapted for compact size and very good in terms power density per unit of volume. However, piezoelectric coupling decreases very fast at micrometric scale and relatively large load impedances are required to reach the optimal working point [8]. Besides, other problems must be considered such as aging, depolarization and brittleness. For low frequency applications, like those related to wearable sensors, polymer-based materials (e.g. dielectric elastomers) constitute a valid alternative to ceramics because of their flexibility, inexpensiveness and

28 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

Electromagnetic technique is simply realized, according to Faraday's law, by coupling a static magnetic field produced by a permanent magnet and a solenoid in relative motion – one of which usually acts as a stator; the other as a mover. These systems show complementary behaviour in terms of frequency bandwidth and optimal load in relation to piezoelectric techniques. They are recommended for lower frequencies (2-20 Hz), small impedance and medium size [10]. Furthermore, their cost is smaller than other solutions. Most of the com‐ mercial solutions are available at centimetre scales because they exhibit higher power density than piezoelectric devices. On the other hand, the integration of electromagnetic harvesters into micro-electro-mechanical-systems (MEMS) results difficult. However, some of these

2 Figure 3. (a) Simple architecture of em-VEH where a moving magnet (mover) oscillates with 3 respect to a fixed coil (stator). (b) Moving magnet across coil arrangement with discrete 4 components [11-13]. (c) Microfabricated em-VEH where a small magnet oscillates towards a

**Figure 3.** (a) Simple architecture of em-VEH where a moving magnet (mover) oscillates with respect to a fixed coil (stator). (b) Moving magnet across coil arrangement with discrete components [11-13]. (c) Microfabricated em-VEH

(c)

V

Moving magnet

Coil

Spring

(a) (b)

6 Electrostatic harvesters basically consist of a variable capacitor in which one electrode is 7 attached to an oscillating mass suspended by beams and the counter electrode is fixed 8 elsewhere in the support structure. When a force is applied to the mass either the dielectric 9 gap or the overlap surface of electrodes varies depending on the moving direction: the first 10 case being referred to as an in-plane gap-closing converter (Figure 4a), while the second case 11 as an overlap varying converter (Figure 4b). As a consequence, the capacitance changes and 12 additional charges occur at the electrodes in order to balance the bias voltage. Hence, during 13 the movement of the proof mass, a current flows through a load shunted to plates. A similar 14 method of fixed bias voltage is that of charge constrained where a constant charge is held

Electrostatic harvesters basically consist of a variable capacitor in which one electrode is attached to an oscillating mass suspended by beams and the counter electrode is fixed elsewhere in the support structure. When a force is applied to the mass either the dielectric gap or the overlap surface of electrodes varies depending on the moving direction: the first case being referred to as an in-plane gap-closing converter (Figure 4a), while the second case as an overlap varying converter (Figure 4b). As a consequence, the capacitance changes and

16 One of the main disadvantages of electrostatic vibration harvesters is the need of an 17 external voltage source in order to be pre-charged. This fact seems to contrast with the goal 18 of energy recycling from the ambient, but makes sense if the source comes from the energy 19 storage associated to the harvester [12]. In that case the generator only needs to be kick-20 started at the beginning of the conversion process. Some designs overcome this problem by

15 into the plates by means of a battery or another capacitor.

21 using electrets to provide the pre-charge bias voltage [14].

5

durability [9].

1

22

5 planar coil [11].

where a small magnet oscillates towards a planar coil [11].

limitations have been overcome to date [11-13].

Running Title

One of the main disadvantages of electrostatic vibration harvesters is the need of an external voltage source in order to be pre-charged. This fact seems to contrast with the goal of energy recycling from the ambient, but makes sense if the source comes from the energy storage associated to the harvester [12]. In that case the generator only needs to be kick-started at the beginning of the conversion process. Some designs overcome this problem by using electrets to provide the pre-charge bias voltage [14].

6 ICT-Energy - Nanoscale Energy Management

2 Figure 4. (a) Schematic of comb gap-closing electrostatic VEH. (b) Example of MEMS in-3 plane overlap electrostatic VEH [15]. **Figure 4.** (a) Schematic of comb gap-closing electrostatic VEH. (b) Example of MEMS in-plane overlap electrostatic VEH [15].

4 Nevertheless, electrostatic technology is very well suited for MEMS manufacturing as

10 quite bulky size. The power density of commercial harvesters ranges from 10 to 300µW/cc

1

5 employs the same elements of micro accelerometers [14]. Moreover, the silicon based MEMS 6 do not have problems of aging as for piezoelectric materials. However the generated power 7 is pretty much small compared to piezoelectric and electromagnetic [16]. 8 Nowadays, vibration energy harvesters are delivered on the real market by some leading 9 companies such as Perpetuum Ltd, Ferro Solution and Mide Volture although they have Nevertheless, electrostatic technology is very well suited for MEMS manufacturing as employs the same elements of micro accelerometers [14]. Moreover, the silicon based MEMS do not have problems of aging as for piezoelectric materials. However the generated power is pretty much small compared to piezoelectric and electromagnetic [16].

11 relative to acceleration levels of 0.01-1g rms. Prototypes of MEMS-based harvesters have 12 been demonstrated by universities and private teams, though they are still at experimental 13 stage. Beeby et al. have implemented a vibration-powered wireless sensor node with 14 embedded micro-electromagnetic generator [3]. Millimetre-sized electrostatic generators 15 were formerly realized by Roundy et al. [11]. Miao implemented a parametric generator for 16 biomedical applications [17] and examples of piezoelectric nano-mechanical generators are 17 also emerging [18]. Mahmood and Basset have successfully built and tested an efficient 18 MEMS-based electrostatic harvester [19, 20]. 19 Nowadays, vibration energy harvesters are delivered on the real market by some leading companies such as Perpetuum Ltd, Ferro Solution and Mide Volture although they have quite bulky size. The power density of commercial harvesters ranges from 10 to 300µW/cc relative to acceleration levels of 0.01-1g rms. Prototypes of MEMS-based harvesters have been dem‐ onstrated by universities and private teams, though they are still at experimental stage. Beeby et al. have implemented a vibration-powered wireless sensor node with embedded microelectromagnetic generator [3]. Millimetre-sized electrostatic generators were formerly realized by Roundy et al. [11]. Miao implemented a parametric generator for biomedical applications [17] and examples of piezoelectric nano-mechanical generators are also emerging [18]. Mahmood and Basset have successfully built and tested an efficient MEMS-based electrostatic harvester [19, 20].

**Figure 5.** Examples of micro vibration energy harvester: (a) electromagentic [16] and (b) (Perpetuum), (c) piezoelectric (Midé), (d) electrostatic [11].

## **3. Linear spring-mass-damper models of VEHs**

Kinetic energy harvesters are divided into two categories: those that utilize direct application of force and those that make use of the inertial force associated to a moving mass *m*. Inertial generators, are preferred to direct-force devices for vibration energy harvesting as they only need one point of attachment to a vibrating structure, thus allowing a simpler miniaturization.

Figure 6 illustrates basic models of (a) direct and (b) inertial force vibration-based generators independent from the conversion technology. In the second case, the driving force *F*(*t*) is equal to −*mÿ*, where the base vibrations are represented as *y*(*t*) and dot stands for the derivative with respect to time. *z*(*t*) is the relative motion between the housing and proof mass, *k* is the spring stiffness, *d* is the parasitic damping, *fe* represents the electrical restoring force due to the transduction mechanism. Finally, the electrical part includes the resistive load *RL* through which flows the generated current *i*.

Williams and Yates early defined a basic technology independent model of micro-electric generator for vibration energy harvesting [16]. In that case the conversion force *fe* being considered as an electrical damping force proportional to the velocity *fe* = −*d*e*ż*. However, the electrical restoring force can in general be a complex function of the mass displacement, velocity and acceleration. In addition, such an approximation does not take into account of the effect of the electrical branch coupled to the mechanical system.

3 Figure 6. Models of non-technology specific (a) direct-force and (b) inertial-force generators. 4 The electrical force *fe* can be in general a complicated function of displacement, velocity or **Figure 6.** Models of non-technology specific (a) direct-force and (b) inertial-force generators. The electrical force *fe* can be in general a complicated function of displacement, velocity or acceleration.

In the following we describe the lumped parameters model of the VEH by including the electrical domain as sketched in Figure 6b. This can be then applied to different types of linear conversion systems. The coupled governing equations of a generic 1-DOF vibration-driven generator are derived by the second Newton's law and Kirchhoff's law as follows 6 Williams and Yates early defined a basic technology independent model of micro-7 electric generator for vibration energy harvesting [16]. In that case the conversion force *fe* 8 being considered as an electrical damping force proportional to the velocity *fe* = *d*e*ż*. 9 However, the electrical restoring force can in general be a complex function of the mass

10 displacement, velocity and acceleration. In addition, such an approximation does not take

13 electrical domain as sketched in Figure 6b. This can be then applied to different types of

$$m\ddot{z} + d\dot{z} + kz + \alpha V = -m\ddot{y}\_{\prime} \tag{1}$$

$$
\dot{V} + (\alpha\_c + \alpha\_i)V = \alpha\_c \lambda \dot{z},
\tag{2}
$$

where dots stand for time-derivatives. The first equation describes the dynamics of the inertial mass while the second equation accounts for the coupled electrical circuit. *V* is the produced voltage across the electrical resistance and *α* is the electromechanical coupling factor; *ω<sup>c</sup>* represents the characteristic cut-off frequencies of the electrical circuit of the system operating as high-pass filter due to the specific transduction technique. This parameter is the inverse to the characteristic time *τ* of the electrical branch, that is *ωc*=1/*τ*. *ω<sup>i</sup>* has the same meaning but corresponding to the internal resistance *Ri* of the system. Finally, *λ* is the electromechanical conversion factor. These characteristic parameters are derived from the harvester design depending on the specific conversion method and architecture as explained in the following examples. Hereafter we will only treat the piezoelectric and electromagnetic case, as the electrostatic is inherently nonlinear when considering the electrical force between close electrostatic plates. 16 *mz dz kz V my* , (1) () , <sup>17</sup>*V Vz ci c* (2) 18 where dots stand for time-derivatives. The first equation describes the dynamics of the 19 inertial mass while the second equation accounts for the coupled electrical circuit. *V* is the 20 produced voltage across the electrical resistance and is the electromechanical coupling 21 factor; *<sup>c</sup>* represents the characteristic cut-off frequencies of the electrical circuit of the 22 system operating as high-pass filter due to the specific transduction technique. This 23 parameter is the inverse to the characteristic time of the electrical branch, that is *c*=1/. *i* 24 has the same meaning but corresponding to the internal resistance *Ri* of the system. Finally, 25 is the electromechanical conversion factor. These characteristic parameters are derived 26 from the harvester design depending on the specific conversion method and architecture as 27 explained in the following examples. Hereafter we will only treat the piezoelectric and

#### **3.1. Case 1: piezoelectric cantilever generator**

1

2

5 acceleration.

**Figure 5.** Examples of micro vibration energy harvester: (a) electromagentic [16] and (b) (Perpetuum), (c) piezoelectric

Kinetic energy harvesters are divided into two categories: those that utilize direct application of force and those that make use of the inertial force associated to a moving mass *m*. Inertial generators, are preferred to direct-force devices for vibration energy harvesting as they only need one point of attachment to a vibrating structure, thus allowing a simpler miniaturization. Figure 6 illustrates basic models of (a) direct and (b) inertial force vibration-based generators independent from the conversion technology. In the second case, the driving force *F*(*t*) is equal to −*mÿ*, where the base vibrations are represented as *y*(*t*) and dot stands for the derivative with respect to time. *z*(*t*) is the relative motion between the housing and proof mass, *k* is the spring stiffness, *d* is the parasitic damping, *fe* represents the electrical restoring force due to the transduction mechanism. Finally, the electrical part includes the resistive load *RL* through

Williams and Yates early defined a basic technology independent model of micro-electric generator for vibration energy harvesting [16]. In that case the conversion force *fe* being considered as an electrical damping force proportional to the velocity *fe* = −*d*e*ż*. However, the electrical restoring force can in general be a complex function of the mass displacement, velocity and acceleration. In addition, such an approximation does not take into account of the

**3. Linear spring-mass-damper models of VEHs**

30 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

effect of the electrical branch coupled to the mechanical system.

(Midé), (d) electrostatic [11].

which flows the generated current *i*.

Starting by the constitutive piezoelectric equations [21], for a bimorph piezoelectric cantilever like the one shown in Figure 2c with active layers wired in parallel, the above characteristic parameters are derived

$$\alpha = kd\_{31} / h\_{p}k\_{2'} \qquad \qquad \text{(a)} \qquad \lambda = \alpha R\_{L'} \tag{b}$$

$$o\_c = 1 / R\_L \mathbb{C}\_{p'} \qquad \qquad \text{(c)} \qquad o\_i = 1 / R\_i \mathbb{C}\_{p'} \qquad \text{(d)} \tag{3}$$

where *hP* and *hs* are the thickness of piezoelectric and support layer respectively; *d*31, *E*p, *ε*0, *ε<sup>r</sup>* and *Cp* are the piezoelectric strain factor, Young's modulus, vacuum and relative dielectric permittivity and, finally, the equivalent capacitance of piezoelectric beam. These constants are related to the following structural parameters as derived in [7]

$$\begin{aligned} k &= k\_1 k\_2 \mathbb{E}\_{p^\*} \\ k\_1 &= \frac{2I}{b(2l\_b + l\_m - l\_e)}, \quad \text{(b)} & \quad k\_2 &= \frac{3b(2l\_b + l\_m - l\_e)}{l\_b^2 \left(2l\_b + \frac{3}{2}l\_m\right)}, \quad \text{(c)}\\ b &= \frac{h\_s + h\_p}{2}, \quad \text{(d)} & \quad I = 2\left[\frac{w\_b h\_p^3}{12} + w\_b h\_p b^2\right] + \frac{E\_s \wedge E\_p w\_b h\_s^3}{12}, \quad \text{(e)} \end{aligned} \tag{4}$$

where *k* represents the effective elastic stiffness, *k*1 and *k*2 are the average strain to mass displacement and the input force to average induced stress, *b* is a geometrical parameter of the bimorph structure and *I* is the composite inertial moment of the beam. Usually, the internal resistance *Ri* of a piezoelectric crystal is very high, hence *ω<sup>i</sup>* is negligible.

#### **3.2. Case 2: electromagnetic generator**

Let consider a simple "magnet in-line coil" electromagnetic generator as schematized in Figure 3a. we can rewrite the characteristic parameters as

$$\begin{aligned} \alpha &= \text{Bl} \mid \text{R}\_{\text{L}'} & \qquad \text{(a)} & \qquad \lambda &= \text{Bl} = \alpha \text{R}\_{\text{L}'} & \qquad \text{(b)}\\ \alpha\_c &= \text{R}\_{\text{L}} \mid \text{L}\_{\text{e}'} & \qquad \text{(c)} & \qquad \alpha\_i &= \text{R}\_i \nmid \text{L}\_{\text{e}'} & \qquad \text{(d)} \end{aligned} \tag{5}$$

with *B* representing the magnetic field across the coil of total length *l* and self-inductance *L*e. Even in this case, by assuming an internal resistance of the coil *R*0 small with respect to the external load (*R*0<< *R*L), we can neglect *ω<sup>i</sup>* . Actually, the above fundamental parameters are also valid for other systems such as "magnet across coil" arrangement depicted in Figure 3b.

In both cases the governing equations are the same, and they can be also rewritten in a more convenient nondimensional form similar to [22].

#### **3.3. Transfer functions**

Let now consider the simple case of harmonic excitation *ÿ*(*t*) = *Y*0ejωt as input. We can transform the motion equations(1) and (2) into the Laplace domain, with *s=jω* as the Laplace variable (where *j* stands for the imaginary unit). The function *Y*(s), *Z*(s), and *V*(s) are the acceleration amplitude, mass displacement and output voltage delivered to the resistive load respectively. Thus, the governing equations for the single-mass generator can be rewritten as the system

31 2 / , (a) , (b) 1 / , (c) 1 / , (d) *p L*

 w

where *hP* and *hs* are the thickness of piezoelectric and support layer respectively; *d*31, *E*p, *ε*0, *ε<sup>r</sup>* and *Cp* are the piezoelectric strain factor, Young's modulus, vacuum and relative dielectric permittivity and, finally, the equivalent capacitance of piezoelectric beam. These constants are

2

3 3 2

2 /

is negligible.

. Actually, the above fundamental parameters are also

(4)

*bb m*

*bme*

ç ÷ + è ø

l a

= = (3)

*c Lp i ip*

*R C R C*

<sup>2</sup> 3 (2 ) , (b) , (c) (2 ) <sup>3</sup> <sup>2</sup>

, (d) 2 , (e) <sup>2</sup> <sup>12</sup> <sup>12</sup>

ê ú ë û

*b p*

where *k* represents the effective elastic stiffness, *k*1 and *k*2 are the average strain to mass displacement and the input force to average induced stress, *b* is a geometrical parameter of the bimorph structure and *I* is the composite inertial moment of the beam. Usually, the internal

Let consider a simple "magnet in-line coil" electromagnetic generator as schematized in Figure

/ , (a) , (b) / , (c) / , (d)

l

 w

with *B* representing the magnetic field across the coil of total length *l* and self-inductance *L*e. Even in this case, by assuming an internal resistance of the coil *R*0 small with respect to the

In both cases the governing equations are the same, and they can be also rewritten in a more

Let now consider the simple case of harmonic excitation *ÿ*(*t*) = *Y*0ejωt as input. We can transform the motion equations(1) and (2) into the Laplace domain, with *s=jω* as the Laplace variable (where *j* stands for the imaginary unit). The function *Y*(s), *Z*(s), and *V*(s) are the acceleration

valid for other systems such as "magnet across coil" arrangement depicted in Figure 3b.

 a

= = (5)

*L L*

*c Le i ie Bl R Bl R R L R L*

= = =

*s p b p s p bs*

*h h w h E Ewh*

*<sup>I</sup> bl l l k k bl l l ll l*

+ é ù = =++ ê ú

of a piezoelectric crystal is very high, hence *ω<sup>i</sup>*

+ - = = + - æ ö

*kd h k R*

= =

32 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

related to the following structural parameters as derived in [7]

1 2

*b I whb*

a

w

, (a)

*bme*

**3.2. Case 2: electromagnetic generator**

a

w

external load (*R*0<< *R*L), we can neglect *ω<sup>i</sup>*

**3.3. Transfer functions**

convenient nondimensional form similar to [22].

3a. we can rewrite the characteristic parameters as

1 2

*k kkE*

=

resistance *Ri*

*p*

$$
\begin{pmatrix} ms^2 + ds + k & a \\ -\lambda a\_c s & s + a\_c \end{pmatrix} \begin{pmatrix} Z \\ V \end{pmatrix} = \begin{pmatrix} -mY \\ 0 \end{pmatrix} \tag{6}
$$

the left-side matrix, that we can name *A*, represents the generalized impedance of the oscil‐ lating system. By means of linear algebraic methods we can easily solve the above system equation, so that the displacement *Z*(s) and output voltage *V*(s) are given by

$$Z = \frac{-mY}{\det A} \text{(s} + o\_{\text{c}}\text{)} = \frac{-mY \cdot \text{(s} + o\_{\text{c}}\text{)}}{m\text{s}^3 + (mo\_{\text{c}} + d)\text{s}^2 + (k + a\lambda o\_{\text{c}} + do\_{\text{c}})\text{s} + ko\_{\text{c}}}\text{)}\tag{7}$$

$$V = \frac{-mY}{\det A} \dot{\lambda} \alpha\_c \mathbf{s} = \frac{-mY \cdot \lambda \alpha\_c \mathbf{s}}{m s^3 + (m \alpha\_c + d) \mathbf{s}^2 + (k + a \lambda \alpha\_c + d \alpha\_c) \mathbf{s} + k \alpha\_c}. \tag{8}$$

Hence, the transfer functions between displacement and voltage over input acceleration are therefore given by

$$H\_{\rm ZY}(\mathbf{s}) = \frac{\mathbf{Z}}{\mathbf{Y}} \prime \quad \text{ (a)} \qquad \qquad H\_{\rm VY}(\mathbf{s}) = \frac{\mathbf{V}}{\mathbf{Y}} \text{.} \tag{9}$$

By substituting *s=jω* in (8), we can calculate the electrical power dissipated across the resistive load

$$P\_{\epsilon}(o) = \frac{\left| V(joo) \right|^{2}}{2R\_{L}} = \frac{\left| H\_{\psi\_{Y}}(joo) \right|^{2} \left| Y(joo) \right|^{2}}{2R\_{L}} = $$

$$= \frac{Y\_{0}^{2}}{2R\_{L}} \left| \frac{-m\lambda o\_{c}joo}{(o\_{c} + jo)(-m o^{2} + jo d + k) + a\lambda o\_{c}joo} \right|^{2} . \tag{10}$$

By introducing the natural frequency of the undamped oscillation *ω<sup>n</sup>* = *k* / *m* and the normal‐ ized damping factor *ζ* =*d* / 2*mωn* into the above equation, it becomes

$$P\_c(o) = \frac{Y\_0^2}{2R\_L} \left| \frac{-\lambda o\_c jo}{(o\_c + jo)(o\_n^2 - o^2 + 2\zeta o\_n jo) + a\lambda o\_c jo \;/\; m} \right|^2 \tag{11}$$

In Figure 7 the graph of the normalized power function over the nondimensional variable *ω*/*ω<sup>n</sup>* of a generic vibration-based generator is shown.

**Figure 7.** Normalized power function of a generic vibration-base harvester.

The main limits of a linear vibration energy harvester include:


In any case, for applications and environments that feature a vibration source consistent in frequency and time, linear systems can still represent an optimal choice.

#### **4. Beyond linear energy harvesting**

In most of the reported studies, the energy harvesters are designed as linear oscillators that match their resonant frequencies to the excitation frequencies of the environment to achieve the maximum output power.

This condition can be easily performed when the excitation frequency is well known and stable in time. It is in fact possible to choose the correct geometry and harvester dimension for frequency matching. However, when the ambient excitation frequency is unknown or varies in time, the previous conditions are not guaranteed. Therefore a harvester with a fixed resonant frequency is not able to achieve an optimal output power.

Various strategies have been investigated to overcome this practical inconvenient and increase the bandwidth of vibration-based harvesters. In the following the state-of-the-art techniques are summarised in three main categories: resonance frequency tuning, multimodal oscillators and frequency up-conversion. A complete technique review is presented by [23].

#### **4.1. Frequency tuning**

In Figure 7 the graph of the normalized power function over the nondimensional variable *ω*/*ω<sup>n</sup>*

z=0.05 z=0.1 z=0.2 z=0.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

<sup>w</sup>/w<sup>n</sup>

**•** narrow bandwidth, that also implies applications with vibrating sources tuned around

**•** at MEMS scale: small inertial mass and maximum displacement, limited power not suitable

In any case, for applications and environments that feature a vibration source consistent in

In most of the reported studies, the energy harvesters are designed as linear oscillators that match their resonant frequencies to the excitation frequencies of the environment to achieve

This condition can be easily performed when the excitation frequency is well known and stable in time. It is in fact possible to choose the correct geometry and harvester dimension for

of a generic vibration-based generator is shown.

34 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

0

resonant frequency of the harvester,

for milliwatts electronics (10-100mW).

**4. Beyond linear energy harvesting**

the maximum output power.

**Figure 7.** Normalized power function of a generic vibration-base harvester.

The main limits of a linear vibration energy harvester include:

**•** versatility and adaptation to variable spectrum vibration sources,

frequency and time, linear systems can still represent an optimal choice.

1

2

3

Power

4

5

6

In some cases it has been demonstrated [24] that the resonance frequency of an oscillator can be tuned to the main exciting frequency in two different ways: passive and active. The passive mode requires an intermittent power input (manual or automatic) to check and tune the system until the frequency match is complete, then the power requirement is zero since the excitation frequency varies again. The active mode is more power demanding since a continuous power is needed to tune the system; this higher power consumption brings the effect to increase the harvester efficiency.

The tuning mechanisms can be realised mechanically using springs or screws, with magnets or using a piezoelectric material.

Few works have been presented in the last years showing possible manual parameter adjust‐ ments to change the harvester stiffness or its mass configuration. The oscillator stiffness is changed with a pre-loaded or a pre-deflected, performing a softening or a hardening of the system.

**Figure 8.** Three tunable vibration-based solutions: (a) Piezoelectric cantilever with a movable mass (source: Wu et al. 2008 [25]), (b) Piezoelectric cantilever with magnetic tuning (source: Challa et al. 2008 [26]), (c) Piezoelectric beam with a scavenging and a tuning part (source: Roundy and Zhang 2005 [27] ).

In Figure 8 three tunable solutions are presented. The three harvesters are realized with a piezoelectric beam, in which the tuning mechanism is purely mechanical in the first case by [25], magnetic in the second by [26], and piezoelectric in the third by [27].

The solution (a) is realized with a proof mass consisting in a fixed part combined with a movable part. The gravity centre of the proof mass can be adjusted by driving the screw. The fixed part of the mass is made with a relatively small density material and the tuning mass with a higher density one. In this way the frequency tunability is increased by moving the distance of the centre of gravity of the proof mass. In this prototype, the frequency range can be manually varied in the 130-180 Hz range moving the tip mass up to 21 mm.

The solution (b) of Figure 8 is realized by coupling two magnets fixed to the free end of the cantilever beam to two other magnets fixed to the top and bottom sides of the enclosure device. All the magnets have a vertical magnetization in such a way to perform an attractive and repulsive force on each side of the beam. Manually tuning the distance between the magnets using a screw, the magnetic force can be changed inducing a change in the cantilever stiffness. The resonant frequency can be varied in the 22-32 Hz range.

The latter case (c), of Figure 8, presented by Roundy and Zhang [25], is an active tuning mechanism. The electrode was etched to create both scavenging and tuning parts on the same beam. They analytically demonstrated that an active tuning actuator never resulted in a net increase in power output. This is explained because the power required to continuously tune the beam resonant frequency exceeds the harvested power increase.

Table 1, taken from Tang et al.[25], summarizes various tuning methods.


**Table 1.** Summary of various resonance tuning methods (source: Tang et al. 2010 [24]).

#### **4.2. Multimodal energy harvesting**

The multimodal approach consists in a multi-vibration harvester, designed to be excited when the natural driving frequency approaches one natural frequency of the harvester. In this case

useful power can be harvested over multiple frequency spectra, increasing the bandwidth that can be covered for efficient energy harvesting.

fixed part of the mass is made with a relatively small density material and the tuning mass with a higher density one. In this way the frequency tunability is increased by moving the distance of the centre of gravity of the proof mass. In this prototype, the frequency range can

The solution (b) of Figure 8 is realized by coupling two magnets fixed to the free end of the cantilever beam to two other magnets fixed to the top and bottom sides of the enclosure device. All the magnets have a vertical magnetization in such a way to perform an attractive and repulsive force on each side of the beam. Manually tuning the distance between the magnets using a screw, the magnetic force can be changed inducing a change in the cantilever stiffness.

The latter case (c), of Figure 8, presented by Roundy and Zhang [25], is an active tuning mechanism. The electrode was etched to create both scavenging and tuning parts on the same beam. They analytically demonstrated that an active tuning actuator never resulted in a net increase in power output. This is explained because the power required to continuously tune

be manually varied in the 130-180 Hz range moving the tip mass up to 21 mm.

36 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

The resonant frequency can be varied in the 22-32 Hz range.

the beam resonant frequency exceeds the harvested power increase.

**Table 1.** Summary of various resonance tuning methods (source: Tang et al. 2010 [24]).

The multimodal approach consists in a multi-vibration harvester, designed to be excited when the natural driving frequency approaches one natural frequency of the harvester. In this case

**4.2. Multimodal energy harvesting**

Table 1, taken from Tang et al.[25], summarizes various tuning methods.

One way to design a multimodal VEH consists in the combination of more transduction mechanisms together. In [24] a hybrid scenario was presented by Tadesse et al. as shown in Figure 9a. The harvester consists of a cantilever beam with piezoelectric crystal plates bonded on it at a fixed distance each other; a permanent magnet is attached at the cantilever tip oscillating within a coil fixed to the housing structure. It this configuration the electromagnetic transducer generates high output power at the cantilever first mode (at 20 Hz), while the piezoelectric transducer generates higher power at the cantilever second mode (at 300 Hz). The combination of the two schemes in one device is able to improve significantly the harvester response covering two frequency ranges. The drawback of this solution is the difficulty in combining the output power from two different mechanisms, thus requiring two separate converting circuits.

A different approach rather than exploiting the energy present at different modes of a single oscillator is to design a cantilever array integrated in one single device. If the geometric parameters of the harvester are appropriately selected, a wide vibration bandwidth can be exploited. In Figure 9b and 9c two different array solutions are presented.

**Figure 9.** Multimodal VEH schematics. (a) Hybrid harvester with piezoelectric and electromagnetic transduction mech‐ anisms (source: Tadesse et al. 2009 [28]). Piezoelectric cantilever arrays with various lengths and tip masses (b) (source: Shahruz 2006 [28]) and same cantilevers with different tip masses (c) (source: Ferrari et al. 2008 [29]).

In the first design, (b), various cantilevers with different lengths and tip masses attached to a common base compose a piezoelectric cantilever array. In the second, (c), the cantilevers are the same but the first mode response is changed varying the tip masses. Each cantilever presents a unique resonant frequency, the combination of which into a single device creates a sort of mechanical band-pass filter. By properly selecting the different parameters, the device can be designed to provide a voltage response on a wide frequency range.

It has been demonstrated by [30] that the improved bandwidth and performance were worth the modest increase in size of the proposed array device. A cantilever array configuration in respect to the hybrid solution, doesn't present the difficulty in combining the output power from the different mechanisms, but requires one rectifier for each cantilever to avoid output cancellation due to the phase difference between the cantilevers.

As a matter of fact, the multimodal approach increases the bandwidth increasing the volume or the weight of the harvester, thus reducing the energy density. Specifically for the cantilever array, only one cantilever or a subset of them are active at the same time generating a certain amount of power while the others are at off-resonance. Hence, knowing the dominant spectrum of the ambient vibrations, the harvester has to be carefully designed to prevent a dramatic efficiency loss.

#### **4.3. Up-conversion energy harvesting**

In many practical situations, the ambient vibrations suitable for energy harvesting including human vibrations, natural events (i.e. wind, seismic motion), common household goods (i.e. fans, fridges, washing machine), automobiles, airplanes, structures etc. present their frequency content below few hundreds hertz.

**Figure 10.** Vibration power spectra. Figure shows acceleration magnitudes (in db/Hz) vs frequency for three different environments.

As an example in Figure 10, three different spectra computed from vibrations taken from a car hood in motion, an operating microwave oven and a running train floor are presented.

All these different sources produce vibrations that are very large in amplitude and spectral characteristics. However, in these cases, like in great part of the cited examples, the ambient vibrations have their energy distributed over a wide frequency spectrum, with significant predominance of low frequency components and frequency tuning or a multimodal approach is not always possible due to geometrical/dynamical constraints. Hence, another frequencyrobust solution for VEH is to 'transfer' the source vibration frequency to the harvester resonance frequency so that useful power can be harnessed in low frequency excitation scenarios.

A typical up-conversion schematic is presented in Figure 11, where the basic principle of such a device is evident [30]. The oscillator with elastic constant *k* has a resonant frequency in a lower region respect to the resonant frequencies of the piezoelectric beams. When the tooth passes and hits the cantilever tips, the cantilevers start oscillating at their natural frequency. Thus the low frequency vibration of the primary vibrating unit (i.e. the oscillating mass *m*) is transferred to the high frequency vibration of the secondary units (i.e. the piezoelectric cantilevers). This provides a robust low frequency harvesting using high frequency structures as transduction elements.

**Figure 11.** Schematic of an up-conversion VEH (source: Rastegar et al. 2006 [31]).

This frequency up-conversion technique was further pursued in few applications like the generators for low and variable speed rotary machineries [31]. It can be implemented com‐ bining different oscillators and interaction mechanisms, like magnetic or electrostatic, main‐ taining its basic principle.

This technique is a way to decouple the exciting frequency and the harvester vibration one, thus the performances are insensitive to the excitation frequency as long as it is less than the resonant frequency of the harvester. The drawback is that the 'first unit' resonance frequency needs to be tuned to the main frequency of the vibration source. In case the exciting vibrations are spread in a wide frequency range, the advantages of this method lose efficiency.

## **5. Non-linear techniques**

As a matter of fact, the multimodal approach increases the bandwidth increasing the volume or the weight of the harvester, thus reducing the energy density. Specifically for the cantilever array, only one cantilever or a subset of them are active at the same time generating a certain amount of power while the others are at off-resonance. Hence, knowing the dominant spectrum of the ambient vibrations, the harvester has to be carefully designed to prevent a

38 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

In many practical situations, the ambient vibrations suitable for energy harvesting including human vibrations, natural events (i.e. wind, seismic motion), common household goods (i.e. fans, fridges, washing machine), automobiles, airplanes, structures etc. present their frequency

**Figure 10.** Vibration power spectra. Figure shows acceleration magnitudes (in db/Hz) vs frequency for three different

As an example in Figure 10, three different spectra computed from vibrations taken from a car hood in motion, an operating microwave oven and a running train floor are presented.

All these different sources produce vibrations that are very large in amplitude and spectral characteristics. However, in these cases, like in great part of the cited examples, the ambient vibrations have their energy distributed over a wide frequency spectrum, with significant predominance of low frequency components and frequency tuning or a multimodal approach is not always possible due to geometrical/dynamical constraints. Hence, another frequencyrobust solution for VEH is to 'transfer' the source vibration frequency to the harvester resonance frequency so that useful power can be harnessed in low frequency excitation

A typical up-conversion schematic is presented in Figure 11, where the basic principle of such a device is evident [30]. The oscillator with elastic constant *k* has a resonant frequency in a lower region respect to the resonant frequencies of the piezoelectric beams. When the tooth passes and hits the cantilever tips, the cantilevers start oscillating at their natural frequency. Thus the low frequency vibration of the primary vibrating unit (i.e. the oscillating mass *m*) is transferred to the high frequency vibration of the secondary units (i.e. the piezoelectric

dramatic efficiency loss.

environments.

scenarios.

**4.3. Up-conversion energy harvesting**

content below few hundreds hertz.

All the above-mentioned strategies to harvest energy from the environment belong to the entire category of linear (or resonant) harvesting techniques. In general, for a generic linear system, even more complicated than a simple cantilever, the transfer function presents one or more peeks corresponding to the resonance frequencies and thus it is effective mainly when the incoming energy is abundant in that frequency region.

Unfortunately this is a serious limitation when it is required to build a vibration energy harvester of small dimension, for at least two main reasons, the first is that, as discussed above, the frequency spectrum of available vibrations instead of being sharply peaked at some frequency is usually very broad. The second reason is that the frequency spectrum of available vibrations is particularly rich in energy in the low frequency range, and it is very difficult, if not impossible, to build small, low-frequency resonant systems.

Based on these considerations it is clear that vibration harvesters inspired by cantilever-like configurations present a number of drawbacks that limit seriously their field of application.

The optimal vibration harvester characteristics for a broadband response can be summarized in the following points:


As we have seen before the search for the best solution in terms of non-resonant systems should start from the potential energy function *U(z)*. In fact *U(z)* plays the role of dynamical energy storage facility (before transduction) for our mechanical oscillator and thus it is here that we should focus our attention.

The best solution in terms of non-resonant systems should start from the potential energy function *U(z)*. In fact equation (1) can be rewritten as:

$$
\alpha m\ddot{z} + d\dot{z} + \frac{d\mathcal{U}(z)}{dz} + \alpha V = -m\ddot{y},
\tag{12}
$$

where *U(z)* plays the role of dynamical energy storage facility (before transduction) for our mechanical oscillator and thus it is here that we should focus our attention.

To replace a linear oscillator with a non-linear one the condition is:

$$
\mathcal{U}I(z) \neq \frac{1}{2}kz^2 \tag{13}
$$

meaning oscillators whose potential energy is not quadratically dependent on the relevant displacement variable. In recent years few possible candidates have been explored [32-42] running from:

$$\text{LU}(z) = a z^{2n} \tag{14}$$

to other more complicated expressions.

For non-linear oscillators it is not possible to define a transfer function, like in paragraph (3.3), and thus a properly defined resonant frequency even if the power spectral density can show one or more well defined peeks for specific values of the frequencies.

In this section, two of these non-linear potential cases, will be briefly addressed.

#### **5.1. Bistable cantilever**

vibrations is particularly rich in energy in the low frequency range, and it is very difficult, if

Based on these considerations it is clear that vibration harvesters inspired by cantilever-like configurations present a number of drawbacks that limit seriously their field of application.

The optimal vibration harvester characteristics for a broadband response can be summarized

**•** Resonant oscillators should be avoided. In fact a resonant oscillator is capable of harvesting energy only in a very narrow band, around its resonant frequency. Non-resonant oscillators

**•** Increase the capability of harvesting energy at low frequency (below few hundred Hz) because this is where most of the ambient energy is available. Due to geometrical constraints

**•** No need for frequency tuning after the initial set-up of the harvester. Frequency tuning is a feature of resonant systems, thus if we move to non-resonant systems this requirement will

As we have seen before the search for the best solution in terms of non-resonant systems should start from the potential energy function *U(z)*. In fact *U(z)* plays the role of dynamical energy storage facility (before transduction) for our mechanical oscillator and thus it is here that we

The best solution in terms of non-resonant systems should start from the potential energy

where *U(z)* plays the role of dynamical energy storage facility (before transduction) for our

&& (12)

*U z kz* ¹ (13)

<sup>2</sup> ( ) *<sup>n</sup> U z az* = (14)

( ) , *dU z mz dz V my dz* && & + + + = a

<sup>1</sup> <sup>2</sup> ( ) <sup>2</sup>

meaning oscillators whose potential energy is not quadratically dependent on the relevant displacement variable. In recent years few possible candidates have been explored [32-42]

mechanical oscillator and thus it is here that we should focus our attention.

To replace a linear oscillator with a non-linear one the condition is:

not impossible, to build small, low-frequency resonant systems.

40 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

a small dimension linear harvester in not feasible.

function *U(z)*. In fact equation (1) can be rewritten as:

in the following points:

have to be taken into account.

be automatically satisfied.

should focus our attention.

running from:

An interesting option for a nonlinear oscillator is to look for a potential energy that is multistable, instead of mono-stable (like the linear case, i.e. the harmonic potential). A particularly simple and instructive example on how to move from the linear (mono-stable) to a possible nonlinear (bi-stable) dynamics is represented by a slightly modified version of the vibration harvester cantilever analysed above (see Figure 12).

**Figure 12.** Schematic of the bi-stable energy harvester considered (source: Vocca et al., 2012 [32]).

This is a common cantilever with a bending piezoelectric layer. At the cantilever tip a small magnet is added. Under the action of the vibrating ground the pendulum oscillates alterna‐ tively bending the piezoelectric layer and thus generating a measurable voltage signal V. The dynamics of the inverted pendulum tip is controlled with the introduction of an external magnet conveniently placed at a certain distance Δ and with polarities opposed to those of the tip magnet. The interaction between the two magnets generates a force dependent from Δ that opposes the elastic restoring force of the bended beam. As a result, the inverted pendulum dynamics can show three different types of behaviours as a function of the distance Δ:


**•** In between the two previous cases it exists an intermediate condition (Δ=Δ0) where the cantilever swings in a more complex way with small oscillations around each of the two equilibrium positions (left and right of the vertical) and large excursions from one to the other.

The nonlinear potential that can be considered in equation (12) is [32]:

$$\text{LII}(z) = \frac{1}{2}k\_e z^2 + \left(Az^2 + B\Lambda^2\right)^{-\frac{3}{2}}\tag{15}$$

with *ke*, *A* and *B* representing constants related to the physical parameters of the cantilever. Clearly when the distance Δ between the magnets grows very large the second term in (15) becomes negligible and the potential tends to the harmonic potential of the linear case, typical of the cantilever harvester.

In Figure 13 the potential *U(z)* for the condition Δ=Δ0 is shown. In this case the potential energy shows clearly two distinct equilibrium points separated by an energy barrier.

**Figure 13.** Potential energy *U(z)* in (0.15) in arbitrary units when (Δ=Δ0).

The overall qualitative behaviour is somehow summarized in Figure 14, where the average power (*P*avg = *V*rms2 */R*) extracted from this vibration harvester is presented as a function of the distance parameter Δ. As it is well evident there is an optimal distance Δ0) where the power peaks to a maximum [33]. Most importantly such a maximum condition is reached in a full nonlinear regime (bistable condition of the potential) resulting larger (at least a factor 4) than the value in the linear condition (far right in Figure 14).

**Figure 14.** Piezoelectric nonlinear vibration harvester mean electrical power (*P*avg=*V*rms2/*R*) as a function of the distance Δ between the two magnets (source: Cottone et al. 2009 [33]]).

#### **5.2. Buckled beam**

**•** In between the two previous cases it exists an intermediate condition (Δ=Δ0) where the cantilever swings in a more complex way with small oscillations around each of the two equilibrium positions (left and right of the vertical) and large excursions from one to the

> 2 22 <sup>2</sup> <sup>1</sup> () ( ) <sup>2</sup> *U z k z Az B <sup>e</sup>*

with *ke*, *A* and *B* representing constants related to the physical parameters of the cantilever. Clearly when the distance Δ between the magnets grows very large the second term in (15) becomes negligible and the potential tends to the harmonic potential of the linear case, typical

In Figure 13 the potential *U(z)* for the condition Δ=Δ0 is shown. In this case the potential energy

The overall qualitative behaviour is somehow summarized in Figure 14, where the average

distance parameter Δ. As it is well evident there is an optimal distance Δ0) where the power peaks to a maximum [33]. Most importantly such a maximum condition is reached in a full nonlinear regime (bistable condition of the potential) resulting larger (at least a factor 4) than

*/R*) extracted from this vibration harvester is presented as a function of the

shows clearly two distinct equilibrium points separated by an energy barrier.

**Figure 13.** Potential energy *U(z)* in (0.15) in arbitrary units when (Δ=Δ0).

the value in the linear condition (far right in Figure 14).

3

= + +D (15)


The nonlinear potential that can be considered in equation (12) is [32]:

42 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

other.

of the cantilever harvester.

power (*P*avg = *V*rms2

Buckled beams represent other possible structures to implement nonlinear VEH. In particular considering a piezoelectric beam clamped on both ends on a base excited vertically, it has demonstrated by Cottone et al.[33], that if the beam is subjected to a compression along his longitudinal axis (see Figure 15) the harvested power increases if the beam is vibrationally excited by a random noise.

**Figure 15.** Schematic of a piezoelectric buckled beam (source: Vocca et al. 2013 [34]).

The equations that link the beam motion to the output voltage across a resistive load RL are the followings [35]:

$$m\ddot{\mathbf{x}} + \gamma \dot{\mathbf{x}} + k\_3 \mathbf{x}^3 + (k\_2 - k\_1 V)\mathbf{x} + k\_0 V = \sigma \xi',\tag{16}$$

$$\frac{1}{2}\mathbf{C}\dot{V} + \frac{V}{R\_L} = -k\_1 \mathbf{x}\dot{\mathbf{x}} + k\_0 \dot{\mathbf{x}} \tag{17}$$

where m, γ, C and RL are the equivalent mass, the viscous parameter, the coupling capacitance and the resistive load respectively. The term k3 is the beam third order stiffness coefficient; k0 and k1 are the piezoelectric coupling terms due to the bending and to the axial strain of the beam respectively. The term k2 =ka - kb∆L is the stiffness parameter, where ka and kb are constants depending on physical parameters of the beam. When the beam is compressed by increasing ∆L (or equivalently the lateral load P), the stiffness becomes negative and the system becomes bistable. The buckled model in equation (16) is valid for small compressions ∆L. The σξ(*t*) represents the vibration force that drives the beam.

It has been demonstrated in [34], that considering as excitation a random force with a Gaussian distribution, zero mean and exponentially auto-correlated, the output power that represents the amount of the energy harvested per second, increases increasing ∆L. In Figure 16 the average electrical power versus the compression length ratio respectively for the experimental (on the left) and numerical models (on the right) are shown for three noise amplitudes.

**Figure 16.** Electrical average power versus relative compression ∆*L/L (%)* for experiment (left column) and numerical simulation (right column). (Source: Cottone et al. 2012, [34]).

Moreover, the piezoelectric beam produces up to an order of magnitude more electric power when it is compressed than in the unbuckled case.

#### **5.3. Comparison of the two methods**

where m, γ, C and RL are the equivalent mass, the viscous parameter, the coupling capacitance and the resistive load respectively. The term k3 is the beam third order stiffness coefficient; k0 and k1 are the piezoelectric coupling terms due to the bending and to the axial strain of the beam respectively. The term k2 =ka - kb∆L is the stiffness parameter, where ka and kb are constants depending on physical parameters of the beam. When the beam is compressed by increasing ∆L (or equivalently the lateral load P), the stiffness becomes negative and the system becomes bistable. The buckled model in equation (16) is valid for small compressions ∆L. The

It has been demonstrated in [34], that considering as excitation a random force with a Gaussian distribution, zero mean and exponentially auto-correlated, the output power that represents the amount of the energy harvested per second, increases increasing ∆L. In Figure 16 the average electrical power versus the compression length ratio respectively for the experimental (on the left) and numerical models (on the right) are shown for three noise amplitudes.

**Figure 16.** Electrical average power versus relative compression ∆*L/L (%)* for experiment (left column) and numerical

Moreover, the piezoelectric beam produces up to an order of magnitude more electric power

simulation (right column). (Source: Cottone et al. 2012, [34]).

when it is compressed than in the unbuckled case.

σξ(*t*) represents the vibration force that drives the beam.

44 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

In Vocca et al. [34] as a comparison between the cantilever nonlinear configuration and the buckled beam, the same piezoelectric element subjected to a fixed vibrating body in both configurations has been simulated. The piezoelectric oscillators output responses obtained as a function of the nonlinear parameter are compared in Figure 17.

**Figure 17.** Comparison between cantilever and buckled beam response as a function of the noise intensity (source: Vocca et al. 2013, [35]).

The cantilever configuration appears to perform better than the beam in all the conditions. Although the buckled beam configuration is an interesting option for an harvester configura‐ tion where the introduction of a magnetic field is undesirable, generally it becomes evident that the cantilever harvester configuration is the best choice once in presence of an exponen‐ tially correlated noise.

#### **6. Conclusions**

In this chapter we have discussed the various strategies developed for vibration energy harvesting. A specific reference to the role of linearity and nonlinearity has been discussed. We have shown that linear resonant systems are clearly limited in their practical applications even if various techniques have been developed to increase the bandwidth. It has been shown that more complex VEHs based on non-linear mechanical oscillators can outperform them in a number of realistic energy harvesting scenarios. In fact, from comparative works that have been recently published it comes out that nonlinear mononstable and bistable structures are the best opinion for enhancing the overall performances and improving the flexibility of vibration powered electronics. [35]

#### **Author details**

Helios Vocca1 and Francesco Cottone2

1 NiPS Laboratory, Department of Physics, University of Perugia, Perugia, INFN Perugia and Wisepower srl, Italy

2 ESIEE Paris, University of Paris Est, Paris, France

#### **References**


[12] P. Wang*, et al.*, "A micro electromagnetic low level vibration energy harvester based on MEMS technology," *Microsystem Technologies,* vol. 15, pp. 941-951, 2009.

**Author details**

and Wisepower srl, Italy

and Francesco Cottone2

46 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

2 ESIEE Paris, University of Paris Est, Paris, France

monitoring," 2011, p. 798109.

1 NiPS Laboratory, Department of Physics, University of Perugia, Perugia, INFN Perugia

[2] N. Gershenfeld*, et al.*, "The Internet of things," *Scientific American,* vol. 291, pp. 76-81,

[3] P. D. Mitcheson*, et al.*, "Energy harvesting from human and machine motion for wire‐ less electronic devices," *Proceedings of the IEEE,* vol. 96, pp. 1457-1486, 2008.

[4] A. Bilbao*, et al.*, "Ultra-low power wireless sensing for long-term structural health

[5] K. Cook-Chennault*, et al.*, "Powering MEMS portable devices—a review of non-re‐ generative and regenerative power supply systems with special emphasis on piezo‐ electric energy harvesting systems," *Smart materials and structures,* vol. 17, p. 043001,

[6] L. Wang and F. Yuan, "Energy harvesting by magnetostrictive material (MsM) for

[7] T. Ikeda, *Fundamentals of Piezoelectricity*. Walton St, Oxford, UK: Oxford University

[8] A. b. E. Lefeuvre, C. Richard, L. Petit, D. Guyomar, "A comparison between several approaches of piezoelectric energy harvesting," *J. Phys. IV France,* vol. 128, pp.

[9] T. G. McKay*, et al.*, "Soft generators using dielectric elastomers," *Applied Physics Let‐*

[10] G. Poulin*, et al.*, "Generation of electrical energy for portable devices Comparative study of an electromagnetic and a piezoelectric system," *Sensors & Actuators: A. Physi‐*

[11] S. P. Beeby*, et al.*, "A micro electromagnetic generator for vibration energy harvest‐

ing," *Journal of Micromechanics and Microengineering,* vol. 17, p. 1257, 2007.

[1] C. S. Raghavendra*, et al.*, *Wireless sensor networks*: Springer, 2006.

powering wireless sensors in SHM," 2007, pp. 18-22.

*ters,* vol. 98, pp. 142903-142903-3, 2011.

*cal,* vol. 116, pp. 461-471, 2004.

Helios Vocca1

**References**

2004.

2008.

Press 1996.

177-186, 2005.


## **Chapter 4**

**Provisional chapter**

## **Thermoelectric Energy Harvesting**

**Thermoelectric Energy Harvesting**

Douglas Paul Additional information is available at the end of the chapter

Douglas Paul

[28] Y. Tadesse*, et al.*, "Multimodal energy harvesting system: piezoelectric and electro‐ magnetic," *Journal of Intelligent Material Systems and Structures,* vol. 20, pp. 625-632,

[29] S. Shahruz, "Design of mechanical band-pass filters for energy scavenging," *Journal of*

[30] M. Ferrari*, et al.*, "Piezoelectric multifrequency energy converter for power harvest‐ ing in autonomous microsystems," *Sensors and Actuators A: Physical,* vol. 142, pp.

[31] J. Rastegar*, et al.*, "Piezoelectric-based power sources for harvesting energy from plat‐

[32] H. Vocca*, et al.*, "Kinetic energy harvesting with bistable oscillators," *Applied Energy,*

[33] F. Cottone*, et al.*, "Nonlinear energy harvesting," *Physical Review Letters,* vol. 102,

[34] F. Cottone*, et al.*, "Piezoelectric buckled beams for random vibration energy harvest‐

[35] H. Vocca*, et al.*, "A comparison between nonlinear cantilever and buckled beam for

[36] L. Gammaitoni*, et al.*, "Nonlinear oscillators for vibration energy harvesting," *Applied*

[37] L. Gammaitoni*, et al.*, "The benefits of noise and nonlinearity: Extracting energy from

[38] M. Ferrari*, et al.*, "Improved energy harvesting from wideband vibrations by nonlin‐

[39] A. Arrieta*, et al.*, "A piezoelectric bistable plate for nonlinear broadband energy har‐

[40] B. Andò*, et al.*, "Nonlinear mechanism in MEMS devices for energy harvesting appli‐ cations," *Journal of Micromechanics and Microengineering,* vol. 20, p. 125020, 2010. [41] D. A. Barton*, et al.*, "Energy harvesting from vibrations with a nonlinear oscillator,"

[42] S. C. Stanton*, et al.*, "Nonlinear dynamics for broadband energy harvesting: Investiga‐ tion of a bistable piezoelectric inertial generator," *Physica D: Nonlinear Phenomena,* vol.

random vibrations," *Chemical Physics,* vol. 375, pp. 435-438, 2010.

ear piezoelectric converters," *Sensors and Actuators A: Physical,* 2010.

vesting," *Applied Physics Letters,* vol. 97, p. 104102, 2010.

*Journal of vibration and acoustics,* vol. 132, 2010.

239, pp. 640-653, 2010.

forms with low frequency vibration," in *Proc. SPIE*, 2006, p. 617101.

*Sound and Vibration,* vol. 292, pp. 987-998, 2006.

48 ICT - Energy - Concepts Towards Zero - Power Information and Communication Technology

ing," *Smart materials and structures,* vol. 21, 2012.

energy harvesting," *European Physical Journal,* 2013.

*Physics Letters,* vol. 94, pp. 164102-164102-3, 2009.

2009.

2012.

2009.

329-335, 2008.

Additional information is available at the end of the chapter 10.5772/57092

http://dx.doi.org/10.5772/57092

## **1. Introduction**

The generation of electrical energy from thermal energy was originally discovered by Thomas Johann Seebeck in 1822 when he first demonstrated that a thermoelectric voltage was produced after providing a temperature difference across two materials. Jean Charles Athanase Peltier then demonstrated in 1834 that the application of a current could be used to pump heat, an effect with great potential for refrigeration. It was not until the 1850s that Lord Kelvin worked out the physics of the Seebeck and Peltier effects attributing the reversible heat flow discovered by Peltier must have an entropy associated with it and the Seebeck coefficient was a measure of the entropy associated with the electric current. Further developments in the theoretical understanding of thermoelectrics required quantum mechanics. The efficiency of the thermoelectric generation process was derived in 1911 by Edmund Altenkirch.

## **2. Fundamental physics**

Before describing and deriving the main thermodynamic properties and equations, it is worthwhile having a brief review of the key parts of physics required for thermoelectrics. One of the first effects that will be used to derive the thermoelectric efficiency is Joule's law of heating. Joule was the first to demonstrate that any current passing through a resistor produces an amount of heat (Fig. 1(a)). Specifically the heat, *Q* (as a power i.e. energy / time) generated by passing a current, *I* though a resistance, *R* is given by Joule's first law

$$Q = I^2 R \tag{1}$$

It should be clear that thermoelectrics have heat being transported through a range of materials and some understanding of the transport of that heat is required. The heat generated by any process will be transported through a material driven by any temperature

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2014 Paul; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Paul, licensee InTech. This is an open access chapter distributed under the terms of the Creative

**Figure 1.** (a) The heat power generated by Joule heating of a resistance, *R* with current, *I* flowing through the resistance. (b) The thermal transport through a rod of area, *A*, length *L* and thermal conductivity *κ* as defined by Fourier's law of thermal transport.

gradients along the material (Fig. 1(b)). This is given by Fourier's law of heat transport which states for a material with area *A* and thermal conductivity *κ* that

$$Q = -\kappa A \nabla T \left( = -\kappa A \frac{T\_c - T\_h}{L} \text{for 1D transport along a length, } L \right) \tag{2}$$

Strictly speaking the heat transport could be in multiple directions in a complex material with a range of different thermal conductivities in different directions but for most thermoelectrics systems, the designs attempt to keep the heat flow simple by using 1 dimensional constructs. The right hand term of equation 2 is the equation for 1D heat transport which is predominantly the one used in thermoelectric devices and modules.

After discussing the generation and transport of heat, we can now discuss the thermoelectric effects. It is far easier to understand the Peltier effect and so this will be discussed before the Seebeck effect. We need to consider two materials in a thermocouple connected between a hot reservoir of temperature, *Th* and a cold reservoir of temperature *Tc* (Fig. 2(a)). To produce the Peltier effect, a current has to be applied and so Fig. 2(a) demonstrate the system with the current being applied to the ends of material 2. The Peltier coefficient, Π, for this system is given by

$$
\Pi = \frac{Q}{I} \tag{3}
$$

The units of Π are the Volt since heat divided by current is Watts divided by Amps. The physics of what is going on is relatively simple. The Peltier coefficient is the heat energy carried by each electron per unit charge and time from the hot reservoir to the cold reservoir.

The Seebeck effect requires a similar circuit to be constructed but this time, the gap in material two is left open circuit (Fig. 2(b)). The open circuit voltage is proportional to

**Figure 2.** (a) The thermocouple system between two heat reservoirs required to demonstrate the Peltier effect (b) The thermocouple system between two heat reservoirs required to demonstrate the Seebeck effect.

(*Th* − *Tc*) = <sup>∆</sup>*<sup>T</sup>* and the constant of proportionality is called the Seebeck coefficient, *<sup>α</sup>*. More generally the Seebeck coefficient is defined as

$$\mathbf{a} = \frac{\mathbf{d}V}{\mathbf{d}T} \tag{4}$$

The units for the Seebeck coefficient are V/K. The Seebeck coefficient is 1/q times the entropy (Q/T) transported with each electron where *q* is the electron charge. Hence the Peltier effect is just each electron in the electrical current transferring an amount of heat from one reservoir to the other i.e. a heat pump.

To calculate the Seebeck or Peltier coefficients from theory requires one to solve the Boltzmann transport equation in the relaxation time approximation. This is beyond the scope of this text but a full derivation can be found in a range of solid state text books including [1]. From this approach, the Seebeck coefficient can be written in terms of the energy of an electron, *E*, Boltzmann's constant, *kB*, the Fermi level in the thermoelectric material, *EF* and the momentum relaxation time, *τ*, as

$$\alpha = -\frac{k\_B}{q} \int \frac{(E - E\_F)}{k\_B T} \frac{\sigma(E)}{\sigma} dE \tag{5}$$

where the electrical conductivity

2 ICT-Energy - Nanoscale Energy Management Concepts

*Q* = −*κA*∇*T*

transport.

is given by

**(a) Q = I2**

**Q = heat (power i.e energy / time)**

**resistance, R**

**I**

which states for a material with area *A* and thermal conductivity *κ* that

<sup>=</sup> <sup>−</sup>*κ<sup>A</sup> Tc* <sup>−</sup> *Th*

predominantly the one used in thermoelectric devices and modules.

**hot side, Th**

**Area, A**

**R (b) Q =** − κ**A** ∇ **T**

**(energy / t) = Q**

**Figure 1.** (a) The heat power generated by Joule heating of a resistance, *R* with current, *I* flowing through the resistance. (b) The thermal transport through a rod of area, *A*, length *L* and thermal conductivity *κ* as defined by Fourier's law of thermal

gradients along the material (Fig. 1(b)). This is given by Fourier's law of heat transport

Strictly speaking the heat transport could be in multiple directions in a complex material with a range of different thermal conductivities in different directions but for most thermoelectrics systems, the designs attempt to keep the heat flow simple by using 1 dimensional constructs. The right hand term of equation 2 is the equation for 1D heat transport which is

After discussing the generation and transport of heat, we can now discuss the thermoelectric effects. It is far easier to understand the Peltier effect and so this will be discussed before the Seebeck effect. We need to consider two materials in a thermocouple connected between a hot reservoir of temperature, *Th* and a cold reservoir of temperature *Tc* (Fig. 2(a)). To produce the Peltier effect, a current has to be applied and so Fig. 2(a) demonstrate the system with the current being applied to the ends of material 2. The Peltier coefficient, Π, for this system

<sup>Π</sup> = *<sup>Q</sup>*

The units of Π are the Volt since heat divided by current is Watts divided by Amps. The physics of what is going on is relatively simple. The Peltier coefficient is the heat energy carried by each electron per unit charge and time from the hot reservoir to the cold reservoir. The Seebeck effect requires a similar circuit to be constructed but this time, the gap in material two is left open circuit (Fig. 2(b)). The open circuit voltage is proportional to

**cold side, Tc**

**L**

*<sup>I</sup>* (3)

(2)

**Q =** −κ**A <sup>T</sup> <sup>c</sup>** <sup>−</sup>**<sup>T</sup> <sup>h</sup>**

*<sup>L</sup>* for 1D transport along a length, *<sup>L</sup>*

**Heat L**

$$
\sigma = \int \sigma(E) dE = q \int \mathbf{g}(E) \mu(E) f(E) (1 - f(E)) dE \tag{6}
$$

The functions are the density of states, *g*(*E*), the carrier mobility, *µ*(*E*) and the Fermi function, *f*(*E*). The best thermoelectric materials are always semiconductors and so the equation for the Seebeck coefficient can be integrated over either the conduction band or the valence band to find the solution for n-type semiconductors and p-type semiconductors respectively. If we only consider electrons in the conduction band for energies above the conduction band edge, *Ec* then we have

$$\mathbf{a} = -\frac{k\_B}{q} \left[ \frac{(E\_c - E\_F)}{k\_B T} + \frac{\int\_0^\infty \frac{(E - E\_c)\sigma(E)\mathbf{d}E}{k\_B T}}{\int\_0^\infty \sigma(E)\mathbf{d}E} \right] \text{ for } E > E\_c \tag{7}$$

This equation is still quite complex and it is difficult to see exactly how to optimise thermoelectric materials. Cutler and Mott [2] realised a more useful form for the Seebeck coefficient. By differentiating the Fermi function it is easy to show that *<sup>f</sup>*(<sup>1</sup> <sup>−</sup> *<sup>f</sup>*) = <sup>−</sup>*kBT* <sup>d</sup>*<sup>f</sup>* d*E* and then expanding *<sup>g</sup>*(*E*)*µ*(*E*) in a Taylor's series around *<sup>E</sup>* = *EF* it can be shown that

$$\alpha = -\left. \frac{\pi^2}{3q} k\_B^2 T \left[ \frac{d \ln(\mu(E)g(E))}{dE} \right] \right|\_{E=E\_F} \tag{8}$$

This equation is only valid for degenerately doped material, that is material that has doping above the Mott criteria and so the Fermi energy is greater than the conduction band edge. The doping density, *nc* given by the Mott criteria is

$$m\_c \approx \left(\frac{0.27}{a\_B^\*}\right)^3 \tag{9}$$

where *a*<sup>∗</sup> *<sup>B</sup>* is the effective Bohr radius given by

$$a\_B^\* = \frac{\epsilon\_0 \epsilon\_r h^2}{\pi m^\* q^2} \tag{10}$$

with *<sup>ǫ</sup>*<sup>0</sup> the permittivity of free space (8.85 <sup>×</sup> <sup>10</sup>−12 Fm<sup>−</sup>1), *<sup>ǫ</sup><sup>r</sup>* the dielectric constant of the semiconductor, *h* is Planck's constant and *m*<sup>∗</sup> is the effective mass of the charge carrier in the semiconductor. In metal-insulator theory this makes the degenerate semiconductor doped above Mott criteria metallic. The Cutler and Mott equation 8 now starts to suggests methods for optimising thermoelectric materials. Materials where the mobility and/or the density of states are varying by large amounts around the Fermi level have high Seebeck coefficients.

Further insights into how to increase the Seebeck coefficient can be found by taking the approach by Ziman [1]. If we ignore energy dependent scattering so that the momentum

10.5772/57092

4 ICT-Energy - Nanoscale Energy Management Concepts

*<sup>α</sup>* = −*kB q*

 

*<sup>α</sup>* <sup>=</sup> <sup>−</sup> *<sup>π</sup>*<sup>2</sup> 3*q k*2 *BT* �

The doping density, *nc* given by the Mott criteria is

*<sup>B</sup>* is the effective Bohr radius given by

(*Ec* − *EF*) *kBT* <sup>+</sup>

*Ec* then we have

where *a*<sup>∗</sup>

The functions are the density of states, *g*(*E*), the carrier mobility, *µ*(*E*) and the Fermi function, *f*(*E*). The best thermoelectric materials are always semiconductors and so the equation for the Seebeck coefficient can be integrated over either the conduction band or the valence band to find the solution for n-type semiconductors and p-type semiconductors respectively. If we only consider electrons in the conduction band for energies above the conduction band edge,

> � ∞

(*E*−*Ec* )*σ*(*E*)d*E kBT*

 

�� � � � *E*=*EF*

*<sup>π</sup>m*∗*q*<sup>2</sup> (10)

for *E* > *Ec* (7)

d*E*

(8)

(9)

*σ*(*E*)d*E*

*d* ln(*µ*(*E*)*g*(*E*)) *dE*

0

� ∞

0

This equation is still quite complex and it is difficult to see exactly how to optimise thermoelectric materials. Cutler and Mott [2] realised a more useful form for the Seebeck coefficient. By differentiating the Fermi function it is easy to show that *<sup>f</sup>*(<sup>1</sup> <sup>−</sup> *<sup>f</sup>*) = <sup>−</sup>*kBT* <sup>d</sup>*<sup>f</sup>*

This equation is only valid for degenerately doped material, that is material that has doping above the Mott criteria and so the Fermi energy is greater than the conduction band edge.

> �0.27 *a*∗ *B*

*<sup>B</sup>* <sup>=</sup> *<sup>ǫ</sup>*0*ǫrh*<sup>2</sup>

with *<sup>ǫ</sup>*<sup>0</sup> the permittivity of free space (8.85 <sup>×</sup> <sup>10</sup>−12 Fm<sup>−</sup>1), *<sup>ǫ</sup><sup>r</sup>* the dielectric constant of the semiconductor, *h* is Planck's constant and *m*<sup>∗</sup> is the effective mass of the charge carrier in the semiconductor. In metal-insulator theory this makes the degenerate semiconductor doped above Mott criteria metallic. The Cutler and Mott equation 8 now starts to suggests methods for optimising thermoelectric materials. Materials where the mobility and/or the density of states are varying by large amounts around the Fermi level have high Seebeck coefficients. Further insights into how to increase the Seebeck coefficient can be found by taking the approach by Ziman [1]. If we ignore energy dependent scattering so that the momentum

�3

*nc* ≈

*a*∗

and then expanding *<sup>g</sup>*(*E*)*µ*(*E*) in a Taylor's series around *<sup>E</sup>* = *EF* it can be shown that

**Figure 3.** (a) The Fermi function demonstrating the carrier occupation as a function of energy for a degenerately doped semiconductor. (b) The first derivative of the Fermi function with respect to energy and also the density of states for the same semiconductor. (c) The product of the terms in (b) times (*E* − *EF* ) which is related to the Seebeck coefficient as detailed in the text. It is the asymmetry between the two red areas above and below the Fermi energy that determines the magnitude of the Seebeck coefficient of the semiconductor.

relaxation time is given by *τ*(*E*) and the electron velocity by *v*(*E*) then the electrical conductivity can be written as

$$
\sigma = \frac{q^2}{3} \int \tau(E) v^2(E) \left[ -g(E) \frac{\mathbf{d}f}{\mathbf{d}E} \right] \,\mathrm{d}E \tag{11}
$$

Zimen then demonstrated that the Seebeck coefficient can be written as

$$\mathfrak{a} = \frac{q^2}{3T\sigma} \int \tau(E) v^2(E) \left[ -g(E) \frac{\mathbf{d}f}{\mathbf{d}E} \right] (E - E\_F) \mathrm{d}E \tag{12}$$

Fig. 3 provides a graphical plot of the terms −*g*(*E*) <sup>d</sup>*<sup>f</sup>* d*E* (*<sup>E</sup>* − *EF*) in this equation. With this approach by Ziman, *σ*, *τ* and *v* are all constant and it is therefore the asymmetry between the value of this term above and below the Fermi level that determines the magnitude of the Seebeck coefficient. This results is important when low dimensional structures are considered as large discontinuities of the density of states can potential provide significant enhancements to the Seebeck coefficient.

William Thomson (Lord Kelvin), realised that the Seebeck coefficient varies with temperature (Fig. 4) and so heat is both absorbed and generated in the thermocouples shown in Fig. 2. The gradient of the heat flux is then given by

$$\frac{dQ}{dx} = \beta I \frac{\mathbf{d}T}{\mathbf{d}x} \tag{13}$$

where *β* is the Thomson coefficient. Kelvin then derived the Kelvin relationships that hold for all materials which are

**Figure 4.** The Thomson coefficient is required as there will be a temperature dependence along any thermoelectric material connected to two heat reservoirs at different temperatures and this produces different Seebeck coefficients along the thermoelectric material. In this diagram the Seebeck coefficient varies along the x-direction i.e. *α* = *α*(*x*)

$$
\Pi = \mathfrak{a}T\tag{14}
$$

$$
\beta = T \frac{\text{d}\alpha}{\text{d}\text{T}} \tag{15}
$$

The Peltier and Thomson coefficients are extremely difficult to measure experimentally but the Seebeck coefficient is relatively easy as it only requires a voltage measurement as a function of ∆*T* across the thermoelectric material. The Kelvin relationships therefore allow the measurement of the Seebeck effect to obtain the Peltier and Thomson coefficients.

#### **3. Thermodynamic efficiency**

We are about to derive the thermodynamical efficiency of thermoelectrics as first demonstrated by Altenkirch in 1911. Before undertaking this, it is sensible to review the Carnot cycle efficiency as this is the maximum efficiency for converting a given amount of thermal energy into work done. It will therefore set a maximum amount for any thermal system and will allow us to determine how much scope there may be for improving thermoelectric materials.

Fig. 5 shows the Carnot cycle where the input work done, *Wcom* to a compressor increases the pressure of the water. This input work done is adiabatic so there is no gain or loss of heat within the complete system. The water flows from the compressor to a furnace where an amount of heat, *Q*<sup>1</sup> is input into the water at constant temperature (i.e. along an isotherm) so that the water is converted from water into dry steam. Therefore all the heat energy is being absorbed as the latent heat in this process of changing water into dry steam. The steam has a larger volume than the liquid water and so the volume increases as shown in Fig. 5(b). This

10.5772/57092

**Figure 5.** (a) The Carnot thermodynamical cycle showing a closed cycle water / steam system where input work, *Wcom* is done of the water / steam by a compressor and the work done as output, *Wt* is that which is the output from a turbine. (b) The temperature-volume phase diagram for the Carnot thermodynamic cycle shown in (a).

increase in volume can be used to turn a turbine and the kinetic energy from the volume expansion can be recovered as work done on the turbine, *Wt*. The temperature is reduced in this process. Then to allow the cycle to start again, the steam has to be condensed into water, and the latent heat, *Q*<sup>2</sup> which is removed at constant temperature can be recovered and reused as an input. The process can then start again.

The efficiency of the Carnot cycle is given by

6 ICT-Energy - Nanoscale Energy Management Concepts

**Hot reservoir Th**

**3. Thermodynamic efficiency**

thermoelectric materials.

**dx**

**I Q**

**Figure 4.** The Thomson coefficient is required as there will be a temperature dependence along any thermoelectric material connected to two heat reservoirs at different temperatures and this produces different Seebeck coefficients along

*<sup>β</sup>* <sup>=</sup> *<sup>T</sup>* <sup>d</sup>*<sup>α</sup>*

The Peltier and Thomson coefficients are extremely difficult to measure experimentally but the Seebeck coefficient is relatively easy as it only requires a voltage measurement as a function of ∆*T* across the thermoelectric material. The Kelvin relationships therefore allow the measurement of the Seebeck effect to obtain the Peltier and Thomson coefficients.

We are about to derive the thermodynamical efficiency of thermoelectrics as first demonstrated by Altenkirch in 1911. Before undertaking this, it is sensible to review the Carnot cycle efficiency as this is the maximum efficiency for converting a given amount of thermal energy into work done. It will therefore set a maximum amount for any thermal system and will allow us to determine how much scope there may be for improving

Fig. 5 shows the Carnot cycle where the input work done, *Wcom* to a compressor increases the pressure of the water. This input work done is adiabatic so there is no gain or loss of heat within the complete system. The water flows from the compressor to a furnace where an amount of heat, *Q*<sup>1</sup> is input into the water at constant temperature (i.e. along an isotherm) so that the water is converted from water into dry steam. Therefore all the heat energy is being absorbed as the latent heat in this process of changing water into dry steam. The steam has a larger volume than the liquid water and so the volume increases as shown in Fig. 5(b). This

the thermoelectric material. In this diagram the Seebeck coefficient varies along the x-direction i.e. *α* = *α*(*x*)

 **is temperature dependent dT**

Π = *αT* (14)

dT (15)

**T**

**Cold reservoir Tc**

**x**

$$\text{Efficiency} = \eta = \frac{\text{net work output}}{\text{net work input}} = \frac{W\_l - W\_{com}}{Q\_1} \tag{16}$$

From the first law of thermodynamics (conservation of energy), we have

$$(Q\_1 - Q\_2) - (W\_t - W\_{com}) = 0\tag{17}$$

and so the efficiency can be rewritten as

$$\eta = \frac{Q\_1 - Q\_2}{Q\_1} = 1 - \frac{Q\_2}{Q\_1} \tag{18}$$

Carnot demonstrated using the temperature versus volume diagram for the Carnot cycle (Fig. 5(b)) that the maximum efficiency is only dependent on the maximum (*Th*) and minimum temperatures (*Tc*) in the cycle and so the maximum efficiency known as the Carnot efficiency becomes

$$
\eta\_c = 1 - \frac{T\_c}{T\_h} \tag{19}
$$

The Carnot efficiency is related to the Kelvin statement of the second law of thermodynamics which states that no system operating in a closed cycle can convert all the heat absorbed from a heat reservoir into the same amount of work. This is just a statement that no thermodynamic heat engine is 100 % efficient. The equivalent Caussius statement is that no process is possible whose sole result is the transfer of heat from a colder to a hotter body when no work is done inside the system. Heat must always flow from a hotter to a colder body. Heat can only be pumped from a colder to a hotter body by undertaking work on the system. Hence the Peltier effect provides one mechanism to pump heat when a current and electrical energy provides the work.

The equation clearly demonstrates that the efficiency can be increased by decreasing *Tc* or increasing *Th*. More correctly the larger <sup>∆</sup>*<sup>T</sup>* = *Th* − *Tc* becomes, the higher the efficiency. Therefore for practical systems, the easiest way to increase the efficiency of any heat engine is to increase the hot reservoir temperature, *Th*.

We have already described how a temperature gradient across a material results in heat conduction through Fourier's law when no electrical current flows in the system. If there is an electrical current in the same direction due to the Seebeck effect then the Peltier effect will attempt to oppose the applied temperature gradient. Therefore when the heat flows through a thermoelectric material between hot and cold reservoirs (see Fig. 1(a)), we have to consider not just the Fourier heat transport but also the Peltier effect.

We therefore need to write: Heat flux per unit area = Peltier term + Fourier term

$$\frac{Q}{A} = \Pi I - \kappa \nabla T \tag{20}$$

From the Kelvin relations we have Π = *αT* and the current density, *J* = *<sup>I</sup> <sup>A</sup>* . Therefore this can be rewritten as

$$Q = \alpha IT - \kappa A \nabla T \tag{21}$$

We will now derive the thermodynamic efficiency of a thermoelectric generator. Before we can generate electricity, we need to build a circuit that can deliver power to a load. Fig. 6(a) shows the basic thermoelectric circuit for generating electricity which consists of an n-type and a p-type semiconductor connected electrically in series and thermally in parallel. The output power is delivered to a load which in this case will just be a resistor, *RL*. For a Peltier cooler, a similar circuit can be designed where the load is replaced by a current source or battery as demonstrated in Fig. 6(b). The thermodynamic efficiency of the thermoelectric generator is given by

10.5772/57092

8 ICT-Energy - Nanoscale Energy Management Concepts

electrical energy provides the work.

can be rewritten as

generator is given by

is to increase the hot reservoir temperature, *Th*.

not just the Fourier heat transport but also the Peltier effect.

*<sup>η</sup><sup>c</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *Tc*

The Carnot efficiency is related to the Kelvin statement of the second law of thermodynamics which states that no system operating in a closed cycle can convert all the heat absorbed from a heat reservoir into the same amount of work. This is just a statement that no thermodynamic heat engine is 100 % efficient. The equivalent Caussius statement is that no process is possible whose sole result is the transfer of heat from a colder to a hotter body when no work is done inside the system. Heat must always flow from a hotter to a colder body. Heat can only be pumped from a colder to a hotter body by undertaking work on the system. Hence the Peltier effect provides one mechanism to pump heat when a current and

The equation clearly demonstrates that the efficiency can be increased by decreasing *Tc* or increasing *Th*. More correctly the larger <sup>∆</sup>*<sup>T</sup>* = *Th* − *Tc* becomes, the higher the efficiency. Therefore for practical systems, the easiest way to increase the efficiency of any heat engine

We have already described how a temperature gradient across a material results in heat conduction through Fourier's law when no electrical current flows in the system. If there is an electrical current in the same direction due to the Seebeck effect then the Peltier effect will attempt to oppose the applied temperature gradient. Therefore when the heat flows through a thermoelectric material between hot and cold reservoirs (see Fig. 1(a)), we have to consider

We will now derive the thermodynamic efficiency of a thermoelectric generator. Before we can generate electricity, we need to build a circuit that can deliver power to a load. Fig. 6(a) shows the basic thermoelectric circuit for generating electricity which consists of an n-type and a p-type semiconductor connected electrically in series and thermally in parallel. The output power is delivered to a load which in this case will just be a resistor, *RL*. For a Peltier cooler, a similar circuit can be designed where the load is replaced by a current source or battery as demonstrated in Fig. 6(b). The thermodynamic efficiency of the thermoelectric

*<sup>A</sup>* <sup>=</sup> <sup>Π</sup>*<sup>J</sup>* <sup>−</sup> *<sup>κ</sup>*∇*<sup>T</sup>* (20)

*Q* = *αIT* − *κA*∇*T* (21)

*<sup>A</sup>* . Therefore this

We therefore need to write: Heat flux per unit area = Peltier term + Fourier term

*Q*

From the Kelvin relations we have Π = *αT* and the current density, *J* = *<sup>I</sup>*

*Th*

(19)

**Figure 6.** (a) A thermoelectric generator delivering power through current to a resistive load connected in electrical series to n- and p-type semiconductor thermoelectric materials. (b) A heat pump (or Peltier cooler) where a current is used to transport heat from the hot reservoir to the cold reservoir. By reversing the current, heat from the cold sink can be pumped by the Peltier effect to the hot sink by undertaking work.

$$\eta = \frac{\text{power supplied to load}}{\text{heat absorbed at hot junction}}\tag{22}$$

The power supplied to the load is just the Joule heating of the load resistor, *RL* which is equal to *I*<sup>2</sup>*RL*. The heat absorbed at the hot junction is the Peltier term plus the heat withdrawn from the hot junction as described above. The Peltier heat is given by <sup>Π</sup>*<sup>I</sup>* = *<sup>α</sup>ITh*. If the resistance of the n-type and p-type semiconductor elements in series is *R*, then the current, *I* flowing in the circuit is just given by Ohm's Law as

$$I = \left(\frac{V}{R\_{total}}\right) = \frac{a(T\_h - T\_c)}{R + R\_L} \tag{23}$$

The heat withdrawn from the hot junction is given by the Fourier term but as there will be Joule heating from the generated current from the Seebeck voltage, some heat will also be generated and returned to the hot junction. It is usual to assume that half of the Joule heat will be transport and half will be returned to the hot junction and so

*Qh* = Fourier - Joule heating + half Joule heating returned (24)

$$\mathbf{k} = \mathbf{x}A(T\_h - T\_c) - I^2\mathbf{R} + \frac{1}{2}I^2\mathbf{R} \tag{25}$$

$$
\dot{\lambda} = \kappa A (T\_h - T\_c) - \frac{1}{2} I^2 R \tag{26}
$$

We can now move to calculate the efficiency by combining these terms and assuming that the power supplied to the load is only through Joule heating to produce

$$\eta = \frac{\text{power supplied to load}}{\text{heat absorbed at hot junction}} \tag{27}$$

$$= \frac{\text{power supplied to load}}{\text{Peliiter} + \text{heat without withinflow hot junction (Q}\_h)}\tag{28}$$

$$\eta = \frac{I^2 R\_L}{aIT\_h + \kappa A(T\_h - T\_c) - \frac{1}{2}I^2 R} \tag{29}$$

To find the maximum efficiency, this equation requires to be solved for <sup>d</sup>*<sup>η</sup>* d *RL R* <sup>=</sup> 0. With a little algebra it can be shown that the solution is

$$\eta\_{\text{max}} = (1 - \frac{T\_c}{T\_h}) \frac{\sqrt{1 + ZT} - 1}{\sqrt{1 + ZT} + \frac{T\_c}{T\_h}} \tag{30}$$

where *T* = <sup>1</sup> <sup>2</sup> (*Th* <sup>+</sup> *Tc*) and the figure of merit for thermoelectrics is defined as

$$ZT = \frac{\alpha^2 \sigma}{\kappa} T \tag{31}$$

Equation 30 has two parts. The first part is just the Carnot efficiency given by (1 − *Tc Th* ). The second part accounts for losses and irreversible processes which reduce as the dimensionless figure of merit, *ZT*, increases in value.

Fig. 7 demonstrates the Carnot efficiency and the maximum thermoelectric efficiencies for different ZTs as a function of ∆*T*. Also included are typically efficiencies for other thermodynamic cycles such as the Rankine and Stirling cycles for different thermal heating schemes. It is clear from this figure that thermoelectrics have significantly less efficiency today that Rankine or Stirling cycle engines. This is certainly true for power generation at the large scale. Below about 100 W, however, the Rankine and Stirling cycles become more difficult to sustain at high efficiencies and thermoelectrics have some advantages. This is mainly due to the fact that fluids effectively become more viscous (lossy) when dimensions are reduced below a certain length scale. A second major advantage of the thermoelectric generators is that they have no moving mechanical parts and are therefore significantly more reliable than Rankine or Stirling cycle engines which have compressors and turbines. Indeed, this is the major reason NASA used radioisotope thermoelectric generators for the Voyager space probes which have been operating for over 34 years and have now left the solar system.

So far we have made the assumption that the electrical and thermal properties of the n-type and p-type semiconductor legs are identical. This is seldom, if ever, the case in real thermoelectric materials. ZT needs to be redefined with the Seebeck coefficients, electrical

10.5772/57092

10 ICT-Energy - Nanoscale Energy Management Concepts

We can now move to calculate the efficiency by combining these terms and assuming that

heat absorbed at hot junction (27)

Peltier + heat withdrawn from hot junction (*Qh*) (28)

1 + *ZT* − 1

1 + *ZT* + *Tc*

*Th*

<sup>2</sup> *<sup>I</sup>*2*<sup>R</sup>* (29)

d *RL R*

*T* (31)

<sup>=</sup> 0. With a

(30)

*Th* ). The

the power supplied to the load is only through Joule heating to produce

*<sup>η</sup>* = power supplied to load

*<sup>α</sup>ITh* + *<sup>κ</sup>A*(*Th* − *Tc*) − <sup>1</sup>

To find the maximum efficiency, this equation requires to be solved for <sup>d</sup>*<sup>η</sup>*

*<sup>η</sup>max* = (<sup>1</sup> <sup>−</sup> *Tc*

*Th* ) √

<sup>2</sup> (*Th* <sup>+</sup> *Tc*) and the figure of merit for thermoelectrics is defined as

*ZT* <sup>=</sup> *<sup>α</sup>*2*<sup>σ</sup> κ*

second part accounts for losses and irreversible processes which reduce as the dimensionless

Fig. 7 demonstrates the Carnot efficiency and the maximum thermoelectric efficiencies for different ZTs as a function of ∆*T*. Also included are typically efficiencies for other thermodynamic cycles such as the Rankine and Stirling cycles for different thermal heating schemes. It is clear from this figure that thermoelectrics have significantly less efficiency today that Rankine or Stirling cycle engines. This is certainly true for power generation at the large scale. Below about 100 W, however, the Rankine and Stirling cycles become more difficult to sustain at high efficiencies and thermoelectrics have some advantages. This is mainly due to the fact that fluids effectively become more viscous (lossy) when dimensions are reduced below a certain length scale. A second major advantage of the thermoelectric generators is that they have no moving mechanical parts and are therefore significantly more reliable than Rankine or Stirling cycle engines which have compressors and turbines. Indeed, this is the major reason NASA used radioisotope thermoelectric generators for the Voyager space probes which have been operating for over 34 years and have now left the solar system. So far we have made the assumption that the electrical and thermal properties of the n-type and p-type semiconductor legs are identical. This is seldom, if ever, the case in real thermoelectric materials. ZT needs to be redefined with the Seebeck coefficients, electrical

Equation 30 has two parts. The first part is just the Carnot efficiency given by (1 − *Tc*

√

<sup>=</sup> *<sup>I</sup>*<sup>2</sup>*RL*

little algebra it can be shown that the solution is

figure of merit, *ZT*, increases in value.

where *T* = <sup>1</sup>

= power supplied to load

**Figure 7.** The thermodynamic efficiency of thermoelectric materials as a function of ZT assuming a cold side temperature of 298 K (25 *<sup>o</sup>*C). Also included is the Carnot efficiency and comparisons to Rankine and Stirling thermodynamic cycles.

conductivities and thermal conductivities defined for both n-type and p-type semiconductors to give

$$ZT = \frac{(\alpha\_p - \alpha\_n)^2 T}{\left[\sqrt{\frac{\kappa\_p}{\sigma\_p}} + \sqrt{\frac{\kappa\_n}{\sigma\_n}}\right]^2} \tag{32}$$

For n-type and p-type semiconductor legs of length (area), *Ln* (*An*) and *Lp* (*Ap*) respectively, ZT is a maximum value when the total resistance of the legs times the thermal conductance is a minimum value. This occurs when

$$\frac{L\_n A\_p}{L\_p A\_n} = \sqrt{\frac{\sigma\_n \kappa\_n}{\sigma\_p \kappa\_p}}\tag{33}$$

Up to this point we have assumed that we have a ∆*T* applied across each of the n- and p-type legs of the thermoelectric module. Since each leg has a given thermal conductivity, only a finite ∆*T* can be sustained across the thermoelectric legs. This therefore sets a maximum ∆*T* that can be sustained due to the thermal conductivity and in terms of ZT it is defined as

$$
\Delta T\_{\text{max}} = \frac{1}{2} Z T\_{\text{c}}^2 \tag{34}
$$

**Figure 8.** Left: The phonon energy dispersion versus wavenumber for Si. Right: schematic diagrams of the optic modes and acoustic modes in terms of the spring model for the lattice bonds between atoms in the crystal for an alloy of Si0.5Ge0.5.

#### **4. Thermal conductivity**

The thermal conductivity is one of the key parameters that many researchers aim to reduce to improve the ZT of a material. The thermal and electrical conductivities in bulk materials are linked as was first demonstrated by Wiedemann and Franz. They made the empirical observation that the thermal conductivity divided by the electrical conductivity times temperature was a constant for all metals. One of the greatest successes of the Drude theory of metals was the explanation of the Wiedemann-Franz law as it is now called. The Drude model assumes that the bulk of the thermal transport in metals is by the conduction electrons. This is now know to be incorrect and the success of the Drude model in explaining the Wiedemann-Franz law was a fortuitous cancelation of two factors of 100. It is now known that the Drude approach of applying the classical gas laws cannot be applied to electron gases in solids. The Wiedemann-Franz law, however, is still correct for degenerately doped bulk semiconductors and metals and provides substantial limitations when trying to optimize thermoelectric materials.

Quantum theory now provides a more complete description of the thermal conductivity which will be described. Phonons are the modes of vibrations of interacting particles in elastic crystal lattices. Phonons are quasi-particles which describe the collective excitations of the lattice (modes of vibrations). Fig. 8 shows the simple semi-classical interpretation where the bonds between atoms in a lattice are considered as springs and then by solving the Helmholtz wave equation for the vibrational modes of these springs throughout the lattice, the energy dispersion curves for the phonons can be calculated. Whilst more accurate description of the phonons requires much more detailed quantum mechanical calculations, this simple picture provides the same overall physical picture of the modes. There are two types of modes: optic and acoustic. The acoustic modes are where the oscillations

10.5772/57092

of neighbours are in phase either in a transverse or longitudinal direction whilst the optics modes have the neighbours oscillating in anti-phase and are therefore at higher energy.

12 ICT-Energy - Nanoscale Energy Management Concepts

**LO**

**LA**

**2**

**4**

**6**

**8**

**Frequency (THz)**

**10**

**12**

**14**

**16**

**0**

**4. Thermal conductivity**

thermoelectric materials.

**60**

**SiGe**

**optic modes - neighbours in antiphase**

**+ –**

**LO**

**TA**

**LA**

**acoustic modes - neighbours in phase**

**NB acoustic phonons transmit most thermal energy**

**50**

**TO TO**

**40**

**30**

**Energy (meV)**

**20**

**TA**

**10**

**0**

**Figure 8.** Left: The phonon energy dispersion versus wavenumber for Si. Right: schematic diagrams of the optic modes and acoustic modes in terms of the spring model for the lattice bonds between atoms in the crystal for an alloy of Si0.5Ge0.5.

The thermal conductivity is one of the key parameters that many researchers aim to reduce to improve the ZT of a material. The thermal and electrical conductivities in bulk materials are linked as was first demonstrated by Wiedemann and Franz. They made the empirical observation that the thermal conductivity divided by the electrical conductivity times temperature was a constant for all metals. One of the greatest successes of the Drude theory of metals was the explanation of the Wiedemann-Franz law as it is now called. The Drude model assumes that the bulk of the thermal transport in metals is by the conduction electrons. This is now know to be incorrect and the success of the Drude model in explaining the Wiedemann-Franz law was a fortuitous cancelation of two factors of 100. It is now known that the Drude approach of applying the classical gas laws cannot be applied to electron gases in solids. The Wiedemann-Franz law, however, is still correct for degenerately doped bulk semiconductors and metals and provides substantial limitations when trying to optimize

Quantum theory now provides a more complete description of the thermal conductivity which will be described. Phonons are the modes of vibrations of interacting particles in elastic crystal lattices. Phonons are quasi-particles which describe the collective excitations of the lattice (modes of vibrations). Fig. 8 shows the simple semi-classical interpretation where the bonds between atoms in a lattice are considered as springs and then by solving the Helmholtz wave equation for the vibrational modes of these springs throughout the lattice, the energy dispersion curves for the phonons can be calculated. Whilst more accurate description of the phonons requires much more detailed quantum mechanical calculations, this simple picture provides the same overall physical picture of the modes. There are two types of modes: optic and acoustic. The acoustic modes are where the oscillations

**X**

**wavenumber**

The total thermal conductivity of a semiconductor or metal can be divided into the electrical contribution, *κel* and the lattice contribution from phonons, *κph*, i.e.

$$
\kappa = \kappa\_{el} + \kappa\_{ph} \tag{35}
$$

For non-degenerate semiconductors (low carrier density), *<sup>κ</sup>el* ≪ *<sup>κ</sup>ph* whilst for degenerate semiconductors (high carrier densities including metals) , *<sup>κ</sup>el* ≫ *<sup>κ</sup>ph*. An ideal thermoelectric material should have both high carrier density and a decoupling of the thermal conductivity with *<sup>κ</sup>el* ≪ *<sup>κ</sup>ph* but Wiedemann-Franz prevents this in bulk semiconductors and metals.

In metals where *κel* dominates the the thermal conductivity, Wiedemann-Franz produces

$$\frac{\sigma T}{\kappa} = \frac{3}{\pi^2} \left( \frac{q}{k\_B} \right)^2 = \frac{1}{L} \tag{36}$$

where *L* is the Lorenz number which is equal to 2.44 × 10−<sup>8</sup> WΩK<sup>−</sup>2. There are a number of examples where the Wiedemann-Franz law breaks down which include pure metals at low temperature, alloys where the small *κel* from high electron scattering results in significant contributions from *κph* and certain low dimensional structures where *κph* can dominate over *κel*.

The lattice contribution is in quantum theory the phonon contribution to the thermal conductivity where phonons are the quantised vibration of the lattice. The phenomenological model using phonon scattering which is now used to calculate the phonon contribution to the thermal conductivity this that first published by Callaway [3]. It assumes that the phonon scattering processes can be represented by frequency-dependent relaxation times and uses a formula derived from the Boltzmann transport equation to calculate the thermal conductivity. The lattice thermal conductivity calculated by this approach is

$$\kappa\_{\rm ph} = \frac{k\_B}{2\pi^2} \left(\frac{k\_B}{\hbar}\right)^3 \int\_0^{\frac{\theta\_B}{T}} \frac{\pi\_\circ(\mathbf{x})\mathbf{x}^4 e^\mathbf{x}}{\upsilon(\mathbf{x})(e^\mathbf{x} - 1)^2} d\mathbf{x} \tag{37}$$

where *<sup>θ</sup><sup>D</sup>* is the Debye temperature, *<sup>x</sup>* <sup>=</sup> *<sup>h</sup>*¯ *<sup>ω</sup> kBT* , ¯*<sup>h</sup>* is Planck's constant divided by 2*π*, *<sup>ω</sup>* is the phonon angular frequency, *τ<sup>c</sup>* is the combined phonon scattering time and *υ* is the phonon velocity. The integral has to include all the acoustic and optical phonon modes although there are particular types of systems where for example transport of optical phonons can be forbidden in small enough nanostructures.

The electrical contribution to the thermal conductivity was derived from the Boltzmann transport equation by Nag [4]. For total electron momentum relaxation times of *τ* for electrons of energy, *E*, the electron contribution to the thermal conductivity is

**Figure 9.** The ZT, Seebeck coefficient, electrical conductivity, thermal conductivity and power factor versus carrier density for a typical semiconductor.

$$\kappa\_{el} = \frac{\sigma}{q^2 T} \left[ \frac{\langle \text{\textperth} \rangle \langle E^2 \pi \rangle - \langle E \pi \rangle^2}{\langle \pi^3 \rangle} \right] \tag{38}$$

The clear result that is not ideal for optimising thermoelectric materials at high carrier densities is that *<sup>κ</sup>el* <sup>∝</sup> *<sup>σ</sup>*.

#### **5. Optimising ZT**

In bulk semiconductor thermoelectric materials, once a material with a particular composition has been chosen, the only real parameter that can be varied to optimise ZT is the doping density. The electrical and thermal parameters in bulk materials are coupled through the Wiedemann-Franz law and so simply by improving one parameter through choosing a better doping does not necessarily result in higher ZT. Fig. 9 shows a schematic diagram of the electrical and thermal properties of the bulk thermoelectric semiconductor material Bi2Te3 as a function of doping density. For this example, the maximum ZT is close to 10<sup>19</sup> cm−<sup>3</sup> whilst the maximum power factor is at 1020 cm<sup>−</sup>3. The figure also demonstrates the coupling of the electrical and thermal conductivities at high carrier densities and the inverse relationship between carrier density and the Seebeck coefficient (*<sup>α</sup>* <sup>∝</sup> *<sup>n</sup>*<sup>−</sup> <sup>2</sup> <sup>3</sup> where *n* is the carrier density).

A comparison of the best n-type and p-type ZT values as a function of temperature is demonstrated in Fig. 10. The solid lines are the ZT values for bulk materials. The majority of the 3D bulk results all have ZT values that are around 1 or less. No bulk material has yet been found with a ZT significantly higher than 1. There have, however, been a number of suggestions to improve ZT by going to lower dimensional structures where the

**Figure 10.** Left: a comparison of ZT for p-type material as a function of temperature (p-Sb2Te3, p-PbTe, p- CeFe4Sb12, p-Yb14MnSb<sup>11</sup> [5], p-Si0.71Ge0.29 [6], 2D p-Bi2Te3/Sb2Te<sup>3</sup> [7], 1D Si [8], 0D p-SiGe [9], p-(GeTe)0.85(AgSbTe)0.15 [10], 0D p-Bi*x*Sb2−*x*Te<sup>3</sup> [11], 0D Mg2Si0.4Sn0.6 [12]). Right: a comparison of ZT for n-type material as a function of temperature (n-Bi2Te3, n-PbTe, n-CoSb<sup>3</sup> [5], n-Si0.7Ge0.3 [6], 0D PbSeTe [13], 0D n-SiGe [14], 0D n-PbSe0.98Te0.02/PbTe [15]).

Wiedemann-Franz rule can break down and quantum effects can be used to optimise the ZT value which will be discussed in the next section.

The first major calculations to demonstrate the advantages of moving to low dimensional structures was that of Mildred Dresselhaus [16]. There are multiple ways that low dimensional structures can enhance the value of ZT.

#### **5.1. Low dimensional structures**

14 ICT-Energy - Nanoscale Energy Management Concepts

α

κ

typical semiconductor.

densities is that *<sup>κ</sup>el* <sup>∝</sup> *<sup>σ</sup>*.

**5. Optimising ZT**

carrier density).

**1019 Carrier density (cm–3)**

**1020 1021 1018 1017**

**Figure 9.** The ZT, Seebeck coefficient, electrical conductivity, thermal conductivity and power factor versus carrier density for a

The clear result that is not ideal for optimising thermoelectric materials at high carrier

In bulk semiconductor thermoelectric materials, once a material with a particular composition has been chosen, the only real parameter that can be varied to optimise ZT is the doping density. The electrical and thermal parameters in bulk materials are coupled through the Wiedemann-Franz law and so simply by improving one parameter through choosing a better doping does not necessarily result in higher ZT. Fig. 9 shows a schematic diagram of the electrical and thermal properties of the bulk thermoelectric semiconductor material Bi2Te3 as a function of doping density. For this example, the maximum ZT is close to 10<sup>19</sup> cm−<sup>3</sup> whilst the maximum power factor is at 1020 cm<sup>−</sup>3. The figure also demonstrates the coupling of the electrical and thermal conductivities at high carrier densities and the inverse

A comparison of the best n-type and p-type ZT values as a function of temperature is demonstrated in Fig. 10. The solid lines are the ZT values for bulk materials. The majority of the 3D bulk results all have ZT values that are around 1 or less. No bulk material has yet been found with a ZT significantly higher than 1. There have, however, been a number of suggestions to improve ZT by going to lower dimensional structures where the

*<sup>κ</sup>el* <sup>=</sup> *<sup>σ</sup> q*2*T* 

relationship between carrier density and the Seebeck coefficient (*<sup>α</sup>* <sup>∝</sup> *<sup>n</sup>*<sup>−</sup> <sup>2</sup>

κ**el**

κ**ph**

�*τ*��*E*2*τ*�−�*Eτ*�<sup>2</sup> �*τ*3�

**ZT**

(38)

<sup>3</sup> where *n* is the

α2 σ

σ

Before demonstrating the potential enhancements that low dimensional structures can bring to the ZT of a thermoelectric material, the definition of lower dimensional samples must be considered. If a sample is made with dimensions of length, *L*, width, *w* and thickness, *t* then the dimensionality of the system and the appropriate transport regime for electrons or phonons is inferred by comparing the sample dimension to the various scattering lengths and characteristics lengths defined below. Care is always required for a sample can be, for example, the 2D in terms of electrical transport but 3D in terms of thermal transport.

In the Drude model, the electrical conductivity is defined in terms of the elastic scattering time, *τ*, the effective mass of the electrons in the material, *m*<sup>∗</sup> and the carrier density, *n* as

$$
\sigma = \frac{nq^2 \pi}{m^\*} \tag{39}
$$

This equation is very simplistic in terms of the mechanisms which determine the electrical conductivity and dependent on the temperature and material, additional transport and scattering mechanisms including disorder, electron-electron interactions, quantum interference or ballistic transport have to be included. Generally the length scale, *lx* associated with a scattering time, *τ<sup>x</sup>* for some scattering process is linked through the diffusion constant, *D* as

$$d\_{\mathbf{x}} = \sqrt{D\tau\_{\mathbf{x}}} \tag{40}$$

In this form the mobility is defined as *µ* = *<sup>q</sup> <sup>m</sup>*<sup>∗</sup> *<sup>τ</sup>* and the Einstein relation relates the mobility to the diffusion constant for an absolute temperature, *T* as

$$
\mu = \frac{qD}{k\_B T} \tag{41}
$$

In most electronic conduction it is only the electrons close to the Fermi level in energy that need to be considered for which the relevant scale is the Fermi wavelength

$$
\lambda\_F = \frac{2\pi}{k\_F} = \sqrt{\frac{2\pi}{n}} = \frac{h}{\sqrt{2m^\*E\_F}}\tag{42}
$$

The elastic scattering length of electrons is defined as the mean free path, ℓ. (Note - the mean free path is strictly defined as the shortest scattering length between all the scattering mechanisms which includes phase coherent scattering, inelastic scattering, electron-electron scattering, but normally the elastic scattering length is the dominant.) The mean free path is defined generally and for lower dimensions as

$$\ell = \upsilon\_F \tau = \frac{\hbar k\_F}{m^\*} \tau \tag{43}$$

$$\ell\_{3D} = \frac{\hbar}{m^\*} (3\pi^2 \frac{n}{g\_v})^{\frac{1}{5}} \frac{\mu m^\*}{q} = \frac{\hbar \mu}{q} (3\pi^2 \frac{n}{g\_v})^{\frac{1}{5}} \tag{44}$$

$$\ell\_{2D} = \frac{\hbar\mu}{q} \sqrt{2\pi \frac{n}{g\_v}}\tag{45}$$

$$\ell\_{1D} = \frac{\hbar \mu}{q} \pi \frac{n}{g\_v} \tag{46}$$

where *gv* is the valley degeneracy of the semiconductor. The above equations have assumed that the electrons have a spin degeneracy of 2 which is only untrue under large magnetic fields sufficiently high to split the spin degeneracy.

Moving to thermal transport there are a number of models which can be considered to determine the appropriate length scales. The first is the thermal length, *LT* defined as the

10.5772/57092

length over which thermal smearing and the associated phase randomization of an electron of the Fermi distribution which produces and energy uncertainty of *kBT*. This is given by

$$L\_T = \sqrt{\frac{D\hbar}{k\_B T}}\tag{47}$$

The phonon group velocity is defined as *∂ω <sup>∂</sup><sup>q</sup>* where *<sup>ω</sup>* is the phonon angular frequency and *<sup>q</sup>* is the phonon wavenumber. The phonon mean free path in the 3D bulk as determined by the Debye theory, which assumes that the phase and group velocity of the phonons are equal, is give by

$$
\Lambda\_{ph} = \frac{\mathfrak{K} \kappa\_{ph}}{\mathbb{C}\_{\overline{v}} \langle v \rangle \rho} \tag{48}
$$

where *Cv* is the specific heat capacity at constant volume, �*υ*� is the average phonon velocity and *ρ* is the density of phonons. Table **??** gives examples of the phonon mean free path in Si and germanium. The Debye theory is by no means unique and the group velocity of phonons is defined in terms of the phonon dispersion relation as demonstrated by Chen [17]. Table **??** provides a comparison between the mean free paths determined by the Debye and dispersion models. There is a significant difference between the two approaches and this is one of the main issues and problems in determining exactly which dimensionality a system is in terms of the thermal transport. In all the 3D cases below, the numbers are larger than the equivalent electron mean free path.


**Table 1**

16 ICT-Energy - Nanoscale Energy Management Concepts

In this form the mobility is defined as *µ* = *<sup>q</sup>*

defined generally and for lower dimensions as

to the diffusion constant for an absolute temperature, *T* as

*D* as

and scattering mechanisms including disorder, electron-electron interactions, quantum interference or ballistic transport have to be included. Generally the length scale, *lx* associated with a scattering time, *τ<sup>x</sup>* for some scattering process is linked through the diffusion constant,

*<sup>µ</sup>* = *qD*

In most electronic conduction it is only the electrons close to the Fermi level in energy that

The elastic scattering length of electrons is defined as the mean free path, ℓ. (Note - the mean free path is strictly defined as the shortest scattering length between all the scattering mechanisms which includes phase coherent scattering, inelastic scattering, electron-electron scattering, but normally the elastic scattering length is the dominant.) The mean free path is

where *gv* is the valley degeneracy of the semiconductor. The above equations have assumed that the electrons have a spin degeneracy of 2 which is only untrue under large magnetic

Moving to thermal transport there are a number of models which can be considered to determine the appropriate length scales. The first is the thermal length, *LT* defined as the

*<sup>n</sup>* <sup>=</sup> *<sup>h</sup>* √<sup>2</sup>*m*∗*EF*

*<sup>q</sup>* <sup>=</sup> *<sup>h</sup>*¯ *<sup>µ</sup>*

*<sup>q</sup>* (3*π*<sup>2</sup> *<sup>n</sup> gv* ) 1

need to be considered for which the relevant scale is the Fermi wavelength

*<sup>λ</sup><sup>F</sup>* <sup>=</sup> <sup>2</sup>*<sup>π</sup> kF* = 2*π*

<sup>ℓ</sup> <sup>=</sup> *vF<sup>τ</sup>* <sup>=</sup> *hk*¯ *<sup>F</sup>*

*<sup>m</sup>*<sup>∗</sup> (3*π*<sup>2</sup> *<sup>n</sup> gv* ) 1 <sup>3</sup> *<sup>µ</sup>m*<sup>∗</sup>

<sup>ℓ</sup>3*<sup>D</sup>* <sup>=</sup> *<sup>h</sup>*¯

<sup>ℓ</sup>2*<sup>D</sup>* <sup>=</sup> *<sup>h</sup>*¯ *<sup>µ</sup> q* 2*π n gv*

<sup>ℓ</sup>1*<sup>D</sup>* <sup>=</sup> *<sup>h</sup>*¯ *<sup>µ</sup> q π n gv*

fields sufficiently high to split the spin degeneracy.

*lx* <sup>=</sup> <sup>√</sup>*Dτ<sup>x</sup>* (40)

*<sup>m</sup>*<sup>∗</sup> *<sup>τ</sup>* and the Einstein relation relates the mobility

*kBT* (41)

*<sup>m</sup>*<sup>∗</sup> *<sup>τ</sup>* (43)

<sup>3</sup> (44)

(42)

(45)

(46)

#### **5.2. Quantum wells**

Before describing how low dimensional systems can improve ZT, we first require to find the energy solutions for a low dimensional systems. The simplest solution to investigate is the 2D quantum well for electrons, (Fig. 11) where the approximation is made that the potential energy for electrons inside the quantum, *V* = 0 and the potential energy outside the well is infinite (*V* = ∞). We require to solve the time independent Schrödinger equation which is

$$-\frac{\hbar^2}{2m^\*}\frac{\mathbf{d}^2\psi(z)}{\mathbf{d}z^2} + V(z)\psi(x) = E\psi(z)\tag{49}$$

**Figure 11.** The quantized energy states in a quantum well of width, *w*.

Outside the quantum well with *V* = ∞, there are no solutions and the electrons are forbidden to occupy regions outside the quantum well. Inside the quantum well *V*(*z*) = 0 and so the Schrödinger equation reduces to the Helmholtz or wave equation

$$-\frac{\hbar^2}{2m^\*}\frac{\mathbf{d}^2\psi(z)}{\mathbf{d}z^2} = E\psi(z)\tag{50}$$

Any travelling wave solution is a valid solution inside the quantum well to this equation so *ψ*(*z*) = *A* sin(*kz*), *ψ*(*x*) = *A* cos(*kz*), *ψ*(*z*) = *Aeikz* and *ψ*(*x*) = *Ae*−*ikz* (where *A* is an amplitude of the wavefunction) and any mixture of these equations are all potential valid trial solutions. The boundary conditions that a solution must adhere to is that at *z* → ∞ then *ψ*(±∞) = 0. For the infinite quantum well in Fig. 11 this requires *ψ*(0) = *ψ*(*w*) = 0 and so the wavefunctions cannot penetrate outside the quantum well. Also *<sup>ψ</sup>*(*z*) and <sup>d</sup>*<sup>ψ</sup>* <sup>d</sup>*<sup>z</sup>* must be continuous between regions. These boundary conditions therefore require that the only possible solution is

$$
\psi(z) = A\_{\hbar} \sin(k\_{\hbar} z) \qquad \text{with} \ k\_{\hbar} = \frac{n\pi}{w} \text{ and } \ n = 1, 2, 3, \dots \tag{51}
$$

If this wavefunction is substituted back into the Schrödinger equation then the solution for the energy is

$$E = \frac{\hbar^2 k\_{\eta}^2}{2m^\*} = \frac{\hbar^2 \pi^2 n^2}{2m^\* w^2} \text{ for } \ n = 1, 2, 3, \dots \tag{52}$$

10.5772/57092

**Figure 12.** The quantized energy states in a quantum well of width, *w*.

The integers *n* = 1, 2, 3, . . . are called quantum numbers and the solutions provide quantized energy levels that restrict the energy of the electrons into subband states indicated by *E*1, *E*2,... *En*. Whilst in any classical system that has a continuous range of energies the ground state can always have zero energy, in this quantized system however, the ground state with *n* = 1 always has energy. This is known as the zero point energy. In semiconductor materials, if two different materials are placed together then a heterostructure results. By placing one material with a lower conduction band between the same material with a higher conduction band, a quantum well is produced as shown in Fig. 11. The electrons move along the xand y-directions of the quantum well and are quantized in the z-direction then the electrons have parabolic energy dispersions along the x- and y-directions and the quantized energy dispersion only along the z-direction as shown in Fig. 12. This results in the complete energy of each electron subband to be

$$E = \frac{\hbar^2 k\_x^2}{2m^\*} + \frac{\hbar^2 k\_y^2}{2m^\*} + \frac{\hbar^2 \pi^2 n^2}{2m^\* w^2} \text{ for } n = 1, 2, 3, \dots \tag{53}$$

#### **5.3. Density of states**

18 ICT-Energy - Nanoscale Energy Management Concepts

**E**

**Figure 11.** The quantized energy states in a quantum well of width, *w*.

possible solution is

the energy is

Schrödinger equation reduces to the Helmholtz or wave equation

<sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup> 2*m*<sup>∗</sup>

*<sup>ψ</sup>*(*z*) = *An* sin(*knz*) with *kn* <sup>=</sup> *<sup>n</sup><sup>π</sup>*

*<sup>E</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*k*<sup>2</sup> *n* <sup>2</sup>*m*<sup>∗</sup> <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*π*2*n*<sup>2</sup> **E1**

**z**

<sup>d</sup>*z*<sup>2</sup> <sup>=</sup> *<sup>E</sup>ψ*(*z*) (50)

and *n* = 1, 2, 3, . . . (51)

<sup>2</sup>*m*∗*w*<sup>2</sup> for *<sup>n</sup>* <sup>=</sup> 1, 2, 3, . . . (52)

<sup>d</sup>*<sup>z</sup>* must

**E2**

**E3**

**Material 2 Material 1 Material 2**

**0 w**

Outside the quantum well with *V* = ∞, there are no solutions and the electrons are forbidden to occupy regions outside the quantum well. Inside the quantum well *V*(*z*) = 0 and so the

d2*ψ*(*z*)

Any travelling wave solution is a valid solution inside the quantum well to this equation so *ψ*(*z*) = *A* sin(*kz*), *ψ*(*x*) = *A* cos(*kz*), *ψ*(*z*) = *Aeikz* and *ψ*(*x*) = *Ae*−*ikz* (where *A* is an amplitude of the wavefunction) and any mixture of these equations are all potential valid trial solutions. The boundary conditions that a solution must adhere to is that at *z* → ∞ then *ψ*(±∞) = 0. For the infinite quantum well in Fig. 11 this requires *ψ*(0) = *ψ*(*w*) = 0 and so the wavefunctions cannot penetrate outside the quantum well. Also *<sup>ψ</sup>*(*z*) and <sup>d</sup>*<sup>ψ</sup>*

be continuous between regions. These boundary conditions therefore require that the only

If this wavefunction is substituted back into the Schrödinger equation then the solution for

*w*

**Ec**

Equations 6 and 8 both use the density of states, *g*(*E*) which describes the number of available states that electrons can occupy in any material system. The density of states is defined as the number of states per energy per unit volume of real space where if *N* is the number of states then

$$\log(E) = \frac{\text{dN}}{\text{dE}}\tag{54}$$

The density of states is therefore counting the number of states between *E* and *E* + d*E* in energy. In **k**-space, the total number of states, *N* is equal to the volume of the sphere or radius **k** first divided by volume occupied by one state and then divided by the volume of real space. Therefore if we have a 3D volume defined by a cube with side length, *L* then the volume of one state in **<sup>k</sup>**-space is <sup>2</sup>*<sup>π</sup> L* 3 . The number of states is given by

$$N = g\_v g\_s \frac{4\pi k^3}{3} \frac{1}{\left(\frac{2\pi}{L}\right)^3} \frac{1}{L^3} \tag{55}$$

$$=g\_v g\_s \frac{4\pi k^3}{3\left(2\pi\right)^3} \tag{56}$$

where the degeneracy of valleys, *gv* and the spin degeneracy *gs* have been added. For direct bandgap semiconductors, *gv* = 1 whilst for indirect bandgap semiconductors the valley degeneracy will be greater than 1. As an example, *gv* = 2 for silicon. The spin degeneracy, *gs* is virtually always 2 for the majority of systems and this only changes for systems with strong magnetic fields. The trick to working out the density of states is to split the derivative in equation 54 into

$$\log(E) = \frac{\text{dN}}{\text{dE}} = \frac{\text{dN}}{\text{dk}} \frac{\text{dk}}{\text{dE}} \tag{57}$$

so that equation 56 becomes

$$\frac{\text{dN}}{\text{d}k} = g\_v g\_s \frac{4\pi k^2}{\left(2\pi\right)^3} \tag{58}$$

The parabolic bands of the effective mass theory provide *<sup>E</sup>* = *<sup>h</sup>*¯ <sup>2</sup> *k*2 <sup>2</sup>*m*<sup>∗</sup> which rearranging gives

$$k = \sqrt{\frac{2m^\*E}{\hbar^2}}\tag{59}$$

Taking the derivative with respect to energy produces

$$\frac{\mathbf{d}k}{\mathbf{d}E} = \sqrt{\frac{2m^\*}{\hbar^2}} \frac{E^{-\frac{1}{2}}}{2} \tag{60}$$

Equations 58 and 60 can now be combined to produce the 3D density of states

10.5772/57092

**Figure 13.** The electron density of states as a function of energy for (a) 3D (b) 2D (c) 1D and (d) 0D semiconductor systems.

$$g\_{\mathfrak{M}}(E) = \frac{g\_v \mathfrak{g}\_s}{4\pi^2} \left(\frac{2m^\*}{\hbar^2}\right)^{\frac{3}{2}} E^{\frac{1}{2}} \tag{61}$$

The same technique can be repeated for 2D systems where instead of a sphere, the 2D

equivalent is a circle in **k**-space. Repeating the technique for the 3D system, the number of states is

$$N\_{2D} = g\_v g\_s \frac{\pi k^2}{3} \frac{1}{\left(\frac{2\pi}{L}\right)^2} \frac{1}{L^2} \tag{62}$$

$$
\xi = g\_v g\_s \frac{\pi k^2}{\left(2\pi\right)^2} \tag{63}
$$

and repeating the techniques above results in the 2D density of states as

$$g\_{2D}(E) = g\_{\upsilon} g\_{\text{s}} \frac{m^\*}{2\pi\hbar^2} \tag{64}$$

This results is for a single subband in a quantum well (see Figs. 11 and 12) and for a heavily doped system as most thermoelectric materials are requires a summation of the density of states over all the subbands. This results in the 2D density of states at an energy, *E* being the sum over all subband below that energy which is

$$\log\_{2D}(E) = \sum\_{i=1}^{n} g\_{\overline{v}} g\_{s} \frac{m^\*}{2\pi\hbar^2} \Theta\left(E - E\_i\right) \tag{65}$$

where Θ is the Heaviside step function.

20 ICT-Energy - Nanoscale Energy Management Concepts

volume of one state in **k**-space is

in equation 54 into

so that equation 56 becomes

radius **k** first divided by volume occupied by one state and then divided by the volume of real space. Therefore if we have a 3D volume defined by a cube with side length, *L* then the

> 4*πk*<sup>3</sup> 3

4*πk*<sup>3</sup> 3 (2*π*)

where the degeneracy of valleys, *gv* and the spin degeneracy *gs* have been added. For direct bandgap semiconductors, *gv* = 1 whilst for indirect bandgap semiconductors the valley degeneracy will be greater than 1. As an example, *gv* = 2 for silicon. The spin degeneracy, *gs* is virtually always 2 for the majority of systems and this only changes for systems with strong magnetic fields. The trick to working out the density of states is to split the derivative

> <sup>d</sup>*<sup>E</sup>* <sup>=</sup> <sup>d</sup>*<sup>N</sup>* d*k*

> > 4*πk*<sup>2</sup> (2*π*)

d*k*

1 2*π L* 3 1

. The number of states is given by

*<sup>L</sup>*<sup>3</sup> (55)

<sup>3</sup> (56)

<sup>d</sup>*<sup>E</sup>* (57)

<sup>3</sup> (58)

*<sup>h</sup>*¯ <sup>2</sup> (59)

<sup>2</sup> (60)

<sup>2</sup>*m*<sup>∗</sup> which rearranging gives

*k*2

2*π L* 3

*N* = *gv gs*

= *gv gs*

*<sup>g</sup>*(*E*) = <sup>d</sup>*<sup>N</sup>*

d*N* <sup>d</sup>*<sup>k</sup>* <sup>=</sup> *gv gs*

> *k* = 2*m*∗*E*

d*k* <sup>d</sup>*<sup>E</sup>* <sup>=</sup>

Equations 58 and 60 can now be combined to produce the 3D density of states

 2*m*<sup>∗</sup> *h*¯ 2

*<sup>E</sup>*<sup>−</sup> <sup>1</sup> 2

The parabolic bands of the effective mass theory provide *<sup>E</sup>* = *<sup>h</sup>*¯ <sup>2</sup>

Taking the derivative with respect to energy produces

The technique can be repeated for 1D and 0D systems and to summarise the density of states as a function of dimension are

**Figure 14.** Left: A cross sectional TEM image of Ge quantum wells with Si0.2Ge0.8 barriers forming a 2D thermoelectric system [18]. The square highlights a dislocation that limits the performance of this material. Middle: A SEM image of etched 50 nm wide nanowires of Ge/Si0.2Ge0.8 material forming 1D thermoelectric systems. Right: A TEM image of a Ge quantum dot grown on a silicon substrate forming a 0D thermoelectric system for scattering phonons.

$$g\_{3D}(E) = \frac{g\_v g\_s}{4\pi^2} \left(\frac{2m^\*}{\hbar^2}\right)^{\frac{3}{2}} E^{\frac{1}{2}}\tag{66}$$

$$\log\_{2D}(E) = \sum\_{i=1}^{n} \lg\_{\mathbb{P}} \frac{m^\*}{2\pi\hbar^2} \Theta\left(E - E\_i\right) \tag{67}$$

$$g\_{1D}(E) = \frac{1}{2\pi\hbar} \sum\_{i=1}^{n} g\_{\overline{v}} g\_{\overline{s}} \sqrt{\frac{m^\*}{2}} \pi\hbar^2 \Theta \left(E - E\_i\right) \tag{68}$$

$$\mathbf{g}\_{0D}(E) = \sum\_{i=1}^{n} \mathbf{g}\_{\upsilon} \mathbf{g}\_{s} \delta \left( E - E\_{i} \right) \tag{69}$$

with *<sup>δ</sup>*(*<sup>E</sup>* − *Ei*) the Dirac delta function.

#### **5.4. Low dimensional enhancements to ZT**

The density of states as a function of energy for 3D, 2D, 1D and 0D are plotted in Fig. 13. Also plotted is the ideal position for the Fermi energy if the Seebeck coefficient is to be maximised using the Cutler and Mott equation

$$\alpha = -\left. \frac{\pi^2}{3q} k\_B^2 T \left[ \frac{d \ln(\mu(E)g(E))}{dE} \right] \right|\_{E=E\_F} \tag{70}$$

By moving to lower dimensional structures, there is a larger asymmetry in the density of states around the Fermi energy and the above equation and the discussions in section 2 indicates that this increases the Seebeck coefficient. Therefore by choosing systems with lower dimensions, the Seebeck coefficient can be enhanced. A number of experimental examples have demonstrated significant improvements to the Seebeck coefficient from reducing the dimensionality from 3D to 2D, 1D or 0D. Examples of the physical structures are shown in Fig. 14. Quantum wells with the transport either parallel or perpendicular to

10.5772/57092

the quantum wells are used for 2D thermoelectric systems (see Fig. 14 left). Nanowires either grown or etched can be used to form 1D thermoelectric systems (Fig. 14 middle) whilst most 0D thermoelectric systems use quantum dots that are aimed at scattering phonons (Fig. 14 right).

22 ICT-Energy - Nanoscale Energy Management Concepts

on a silicon substrate forming a 0D thermoelectric system for scattering phonons.

*<sup>g</sup>*3*D*(*E*) = *gv gs*

*<sup>g</sup>*2*D*(*E*) =

*<sup>g</sup>*0*D*(*E*) =

with *<sup>δ</sup>*(*<sup>E</sup>* − *Ei*) the Dirac delta function.

**5.4. Low dimensional enhancements to ZT**

maximised using the Cutler and Mott equation

*<sup>g</sup>*1*D*(*E*) = <sup>1</sup>

*<sup>α</sup>* <sup>=</sup> <sup>−</sup> *<sup>π</sup>*<sup>2</sup> 3*q k*2 *BT* 

4*π*<sup>2</sup>

2*πh*¯

*n* ∑ *i*=1

*n* ∑ *i*=1 *gv gs*

2*m*<sup>∗</sup> *h*¯ 2

*n* ∑ *i*=1 *gv gs*

*m*∗

The density of states as a function of energy for 3D, 2D, 1D and 0D are plotted in Fig. 13. Also plotted is the ideal position for the Fermi energy if the Seebeck coefficient is to be

By moving to lower dimensional structures, there is a larger asymmetry in the density of states around the Fermi energy and the above equation and the discussions in section 2 indicates that this increases the Seebeck coefficient. Therefore by choosing systems with lower dimensions, the Seebeck coefficient can be enhanced. A number of experimental examples have demonstrated significant improvements to the Seebeck coefficient from reducing the dimensionality from 3D to 2D, 1D or 0D. Examples of the physical structures are shown in Fig. 14. Quantum wells with the transport either parallel or perpendicular to

*d* ln(*µ*(*E*)*g*(*E*)) *dE*

 *E*=*EF*

3 2 *E*1

*m*<sup>∗</sup>

**5 nm**

(70)

**Si**

**Ge**

<sup>2</sup> (66)

<sup>2</sup> *<sup>π</sup>h*¯ <sup>2</sup><sup>Θ</sup> (*<sup>E</sup>* <sup>−</sup> *Ei*) (68)

<sup>2</sup>*πh*¯ <sup>2</sup> <sup>Θ</sup> (*<sup>E</sup>* <sup>−</sup> *Ei*) (67)

*gv gs<sup>δ</sup>* (*<sup>E</sup>* − *Ei*) (69)

**Figure 14.** Left: A cross sectional TEM image of Ge quantum wells with Si0.2Ge0.8 barriers forming a 2D thermoelectric system [18]. The square highlights a dislocation that limits the performance of this material. Middle: A SEM image of etched 50 nm wide nanowires of Ge/Si0.2Ge0.8 material forming 1D thermoelectric systems. Right: A TEM image of a Ge quantum dot grown 2D quantum well systems can also enhance the electrical conductivity by a number of techniques. If the electrical and thermal transport is along quantum wells then modulation doping can be used to enhance the electrical conductivity. Here the dopants are only placed in the barriers which are at higher energy than the quantum well. The carriers fall into the quantum well and are therefore remote from the ionised dopants that created the carriers. By separating the carriers from the dopants, the Coulomb scattering is reduced and the mobility and electrical conductivity increases. This occurs along with the Seebeck enhancement described above and so higher power factors can be created. The potential disadvantage of this technique is that the electrical conductivity can be so large that it also can increase the thermal conductivity in a detrimental way to the ZT. By optimising the parameters, higher ZTs can be produced.

The third approach to optimising ZT is to reduce the thermal conductivity. There are many ways to reduce the thermal conductivity by adding scattering centres or rough interfaces that scatter phonons. The main issue is to aim for a scattering technique that scatters phonons more than electrons. This is not as easy as it sounds. Adding 0D nanoparticles or quantum dots into a material has been successful at reducing *κ* faster than *σ* as shown in Fig. 10 in a number of material systems for both n- and p-type semiconductors [13, 15]. The quantum dots when of the correct size can scatter phonons much more easily than electrons especially for highly doped samples where the electron mean free paths are typically longer than phonon mean free paths. There are a number of examples of materials such as skutterudites where heavy atoms are inserted to fill voids in the lattice to have a similar effect at the microscale of the lattice [5].

2D superlattices with the electron transport perpendicular to the quantum well and barriers are also good at scattering phonons. The disadvantage of this type of superlattice is that the electrons or holes must quantum mechanically tunnel through the barriers which also significantly reduces the electrical conductivity to typically 3 to 4 times lower than bulk material. The lower thermal conductivity combined with the higher Seebeck from the 2D quantum wells does produce significant enhancements to ZT as show in Fig. 10 [7].

Finally 1D nanowires have also demonstrated substantial improvements to ZT. Boukai et al., [8] demonstrated 10 nm wide Si nanowires which demonstrate enhanced Seebeck coefficients and significantly reduced thermal conductivities compared to bulk Si. The thermal conductivity demonstrated the largest changes with reductions of up to 150 times that of bulk silicon whilst the Seebeck improves by a factor of 2. ZTs at room temperature of 0.3 have been achieved with higher values at lower temperatures. The nanowires demonstrate how confining phonons in low dimensional structures can make significant changes to the ZT of a material.

### **6. The power output from thermoelectric modules**

To this point the majority of this review has concentrated on ways to optimise ZT but for applications it is the current and voltage i.e. the output power produced that is the most

**Figure 15.** A schematic diagram of a complete module with leg length, *L* and contact length, *lc* .

important parameter for applications. Therefore it is important to understand the issues of the output power and how it may be optimised. We will start by considering a complete module as shown in Fig. 15. As the Seebeck coefficient of most materials is ≪ 1 mV/K and most applications will require at least 1 V and in many cases multiple Volts, a large number of legs must be connected in series to achieve suitable output voltage for the applications. If we have *N* legs each of length, *L*, with thermal conductivity, *κ* and contacts of length, *lc* with thermal conductivity, *κ<sup>c</sup>* then the voltage produced is given by [19]

$$V = \frac{\mathfrak{a}N\Delta T}{1 + 2\frac{\kappa l\_c}{\kappa\_c L}}\tag{71}$$

The current can also be calculated and for legs of area *A* and electrical conductivity, *σ* with specific contact resistivity, *ρc*, the current is

$$I = \frac{a\sigma A\Delta T}{2(\rho\_c\sigma + L)(1 + 2\frac{\kappa l\_c}{\kappa\_c L})} \tag{72}$$

By multiplying the voltage and current together, the resulting power is

$$P = \frac{\alpha^2 \sigma A N \Delta T^2}{2(\rho\_c \sigma + L)(1 + 2\frac{\kappa l\_c}{\kappa\_c L})^2} \tag{73}$$

There are a number of issues that the power equation highlights. The first is that the power is dominated by *α*2*σ* which is called the power factor. The second is that the power is proportional to the area of the legs and the number of the legs in the module. The power is also proportional to the square of ∆*T*. Finally, whilst shortening the length of the legs in the module to first order will increase the power, equation 73 demonstrates that as the leg length reduces, the contact resistance of each leg plays a larger part in reducing the output power. For the microfabricated modules, having a low specific contact resistivity can be as important as a material with high ZT and power factor in being able to achieve a high output power.

**Figure 16.** The power as calculated from equation 73 for a module with 525 legs of area 500 *µ*m × 50 *µ*m, leg length of *L* = 20 *µ*m and *lc* = 10 nm.

Another issue is that for many applications the amount of heat may vary and is not constant. Therefore power conditioning along with electrical impedance matching is required so that the thermoelectric module can always operate at the maximum power point. Maximum power point tracking systems are required which automatically adjust the load impedance so that it is always matched to the thermoelectric for all levels of heat and ∆T being applied across the thermoelectric.


**Table 2**

(71)

*<sup>κ</sup>cL* ) (72)

*<sup>κ</sup>cL* )<sup>2</sup> (73)

24 ICT-Energy - Nanoscale Energy Management Concepts

**l c**

**l c**

specific contact resistivity, *ρc*, the current is

power.

**L n ppp n n**

**Figure 15.** A schematic diagram of a complete module with leg length, *L* and contact length, *lc* .

thermal conductivity, *κ<sup>c</sup>* then the voltage produced is given by [19]

By multiplying the voltage and current together, the resulting power is

**hot side electrical insulator**

**Qtransmitted Qrejected**

**Hot sink**

**cold side electrical insulator**

**Cold sink**

important parameter for applications. Therefore it is important to understand the issues of the output power and how it may be optimised. We will start by considering a complete module as shown in Fig. 15. As the Seebeck coefficient of most materials is ≪ 1 mV/K and most applications will require at least 1 V and in many cases multiple Volts, a large number of legs must be connected in series to achieve suitable output voltage for the applications. If we have *N* legs each of length, *L*, with thermal conductivity, *κ* and contacts of length, *lc* with

> *<sup>V</sup>* <sup>=</sup> *<sup>α</sup>N*∆*<sup>T</sup>* <sup>1</sup> + <sup>2</sup> *<sup>κ</sup>lc κcL*

The current can also be calculated and for legs of area *A* and electrical conductivity, *σ* with

<sup>2</sup>(*ρc<sup>σ</sup>* <sup>+</sup> *<sup>L</sup>*)(<sup>1</sup> <sup>+</sup> <sup>2</sup> *<sup>κ</sup>lc*

*<sup>I</sup>* <sup>=</sup> *ασA*∆*<sup>T</sup>*

*<sup>P</sup>* <sup>=</sup> *<sup>α</sup>*2*σAN*∆*T*<sup>2</sup> <sup>2</sup>(*ρc<sup>σ</sup>* <sup>+</sup> *<sup>L</sup>*)(<sup>1</sup> <sup>+</sup> <sup>2</sup> *<sup>κ</sup>lc*

There are a number of issues that the power equation highlights. The first is that the power is dominated by *α*2*σ* which is called the power factor. The second is that the power is proportional to the area of the legs and the number of the legs in the module. The power is also proportional to the square of ∆*T*. Finally, whilst shortening the length of the legs in the module to first order will increase the power, equation 73 demonstrates that as the leg length reduces, the contact resistance of each leg plays a larger part in reducing the output power. For the microfabricated modules, having a low specific contact resistivity can be as important as a material with high ZT and power factor in being able to achieve a high output

#### **7. Applications**

The first thermoelectric application was to power satellites for space applications in 1961 [20]. Space systems use radioisotope thermoelectric generators (RTGs) where a radioactive material heated by the decay and emission of radiation is used as the hot source with a thermoelectric generator to turn the heat into electricity. Plutonium 238 was the main power source used by NASA in most of their 28 RTG systems which operates with temperatures up to 1000 *<sup>o</sup>*C whilst the outside of the spacecraft is used with heat exchangers to provide the cold sink. With such high temperatures, SiGe has been the main thermoelectric material

**Figure 17.** Left: a telecoms laser on a microfabricated Peltier cooler produced by Micropelt. Right: a thermoelectric generator produced by Micropelt showing the cold sink aimed at dissipating heat through air cooling. Copyright Micropelt [21].

used for these generators and the efficiencies can be as high as 6.6 % mainly due to the high ∆T. Both Voyager space missions which are now outside the solar system are powered by SiGe RTGs weighing 37.7 kg and provided 470 W on launch at 40 V. Over a period of time the temperature of the plutonium reduces as a function of the half-life of the radioactive decay so the systems are now generating less than 350 W, 34 years after launch. This is one of the best demonstrations of the robustness of the thermoelectric generator as for such space systems now over a light year away from earth, the power sources must be fit and forget. One problem is that there are no present sources of plutonium 238 as all the weapons nuclear reactors have been shut down and so there are a number of research programmes aiming to develop new RTG technology using available radio isotopes.

The major application for thermoelectric devices at present is as Peltier coolers (Fig. 17 left) to maintain a low temperature for electronic and optoelectronic components such as telecoms lasers and rf sources. It is therefore in the use as a cooler rather than generating electricity that thermoelectrics are predominantly used at present. A number of companies are offering thermoelectric generator demonstrator kits as products (Fig. 17 right) to allow companies to test thermoelectrics for specific applications. The major present applications are for generating electricity to run sensors in a range of predominantly industrial environments where ∆Ts between 20 and 100 *<sup>o</sup>*C can provide sufficient energy for the sensors. The majority of the energy use in the sensors is for the wireless communication that for mobile phone system communication will require powers around 5 mW while many of the sensing elements only require sub-*µ*W powers for making a measurement. These wireless industrial sensing applications are widespread and the cost of the thermoelectric device becomes economical for systems where no installation of wires for communication and/or long term power. It is the large cost of replacing batteries (mainly labour costs) that allows thermoelectrics and other forms of energy harvesting to be cost effect. Most wireless sensors systems now require between 1 and 5 mW of power to run mainly dependent on the distance for the communication and so a square cm area thermoelectric device requires around 50 *<sup>o</sup>*C to provide sufficient power. As the communications consumes the most power, most systems have rechargable battery or super-capacitor storage systems and then use burst modes of communication so that information is only sent when required to save power.

Research results have been published on clothing with integrated thermoelectrics [22]. Only small ∆Ts can be provided across clothing using the air temperature outside the

26 ICT-Energy - Nanoscale Energy Management Concepts

**Figure 17.** Left: a telecoms laser on a microfabricated Peltier cooler produced by Micropelt. Right: a thermoelectric generator produced by Micropelt showing the cold sink aimed at dissipating heat through air cooling. Copyright Micropelt [21].

used for these generators and the efficiencies can be as high as 6.6 % mainly due to the high ∆T. Both Voyager space missions which are now outside the solar system are powered by SiGe RTGs weighing 37.7 kg and provided 470 W on launch at 40 V. Over a period of time the temperature of the plutonium reduces as a function of the half-life of the radioactive decay so the systems are now generating less than 350 W, 34 years after launch. This is one of the best demonstrations of the robustness of the thermoelectric generator as for such space systems now over a light year away from earth, the power sources must be fit and forget. One problem is that there are no present sources of plutonium 238 as all the weapons nuclear reactors have been shut down and so there are a number of research programmes

The major application for thermoelectric devices at present is as Peltier coolers (Fig. 17 left) to maintain a low temperature for electronic and optoelectronic components such as telecoms lasers and rf sources. It is therefore in the use as a cooler rather than generating electricity that thermoelectrics are predominantly used at present. A number of companies are offering thermoelectric generator demonstrator kits as products (Fig. 17 right) to allow companies to test thermoelectrics for specific applications. The major present applications are for generating electricity to run sensors in a range of predominantly industrial environments where ∆Ts between 20 and 100 *<sup>o</sup>*C can provide sufficient energy for the sensors. The majority of the energy use in the sensors is for the wireless communication that for mobile phone system communication will require powers around 5 mW while many of the sensing elements only require sub-*µ*W powers for making a measurement. These wireless industrial sensing applications are widespread and the cost of the thermoelectric device becomes economical for systems where no installation of wires for communication and/or long term power. It is the large cost of replacing batteries (mainly labour costs) that allows thermoelectrics and other forms of energy harvesting to be cost effect. Most wireless sensors systems now require between 1 and 5 mW of power to run mainly dependent on the distance for the communication and so a square cm area thermoelectric device requires around 50 *<sup>o</sup>*C to provide sufficient power. As the communications consumes the most power, most systems have rechargable battery or super-capacitor storage systems and then use burst modes of

aiming to develop new RTG technology using available radio isotopes.

communication so that information is only sent when required to save power.

Research results have been published on clothing with integrated thermoelectrics [22]. Only small ∆Ts can be provided across clothing using the air temperature outside the

**Figure 18.** A schematic diagram of how the energy from a combustion engine in a car is distributed. 25% of the energy produces motion and through the alternator generates electricity to power accessories including the electrics, the air conditioning and the hifi system. 75% of the energy from the fuel is lost mostly through friction and heat. 40% of the fuel energy disappears through the exhaust system hence there is interest in using thermoelectrics to harvest some of this waste energy.

body to provide the cold side temperature. There is enormous interest in such power sources for autonomous wireless health monitoring systems that can be fit and forget. Electrocardiography systems integrated into the clothing and powered by thermoelectrics have already been demonstrated and even tested through cycles of washing in washing machines to check the robustness of the technology. For this technology to be practical, batteries or super capacitors and power management must also be integrated along with the sensors, some processing and communication technology. Whilst the low powers available with the small ∆Ts may struggle to power the present communication systems, new short range communication protocols being developed at present for autonomous sensors by the IEEE are required before this type of application can be aggressively pursued.

The major driver for improved thermoelectrics at present is probably the car industry where European legislation to improve fuel efficiency is driving thermoelectrics research to replace the alternator. The car is an excellent system where thermoelectrics could play a big role as 75% of the fuel ends up as waste heat and the 40 % of waste heat that goes down the exhaust pipe into an environment that could be used to capture this heat and convert it into electricity (Fig. 18) [23, 24]. The temperature of the exhaust system can range from room temperature up to 750 *<sup>o</sup>*C so this is driving work on new thermoelectric materials to replace the best at present which is PbTe with toxic Pb that cannot be used for applications. Initial modelling has suggested that up to a 5% in fuel consumption could be achieved with suitable thermoelectrics with ZT of 1 but the key issue is getting the whole thermoelectric system cheap enough for the market. Also no thermoelectric provides ZT of 1 from room temperature to 750 *<sup>o</sup>*C so segmented modules and/or new materials are required. Most of the major car companies are now working heavily of thermoelectrics and it is only a mater of time before automotive systems become available.

A developing application is thermal photovoltaics. Concentrator photovoltaics are now producing efficiencies of around 45 % when the light is concentrated up to a thousand times. These systems are aimed at solar farms which in sunny regions of the earth have the potential to produce power stations at the 100s of MW scale. Such concentration, however, results in the photovoltaic cells being heating to very high temperatures that results in large thermal cycling which ultimately produces failures reducing the lifetime of the systems. By integrating a thermoelectric with the photovoltaic, not only can electricity be generated from the heat thereby providing a joint and therefore improved system efficiency closer to the Carnot limit but more importantly, the thermoelectric cools the photovoltaic reducing the thermal cycling extremes and increases the lifetime of the system. This increase in lifetime results in substantially cheaper cost per Watt which is the major driver for photovoltaics. It is clear this will be a developing application for thermoelectrics in the future.

At present thermoelectrics requires a "killer application" before volume manufacture will result in widespread use. The automotive applications at present appear to be the major application driver, more by legislation rather than market but it is clear that the potential requirements for thermoelectric generators will improve as fossil fuel prices increase in the future. The autonomous sensor market may well also drive thermoelectrics but the real problem that must be solved is to find sustainable thermoelectric materials. Tellurium is the 9*th* rarest element on earth predicted to run out before 2020 with the present use and so a key requirement for future thermoelectrics is that Te-free materials can be found that can be cheaply produced and with a high ZT and power factor. There is presently an enormous amount of research in thermoelectrics especially aiming to find Te-free materials and to produce modules for higher temperature applications such as the automotive energy harvesting. For students starting in a research career, this is one research area that is likely to expand over the next decades.

#### **Author details**

Douglas Paul

University of Glasgow, School of Engineering, Rankine Building, U.K.

#### **References**


10.5772/57092

[6] J.P. Dismukes, E. Ekstrom, D.S. Beers, E.F. Steigmeier, and I. Kudman. Thermal + electrical properties of heavily doped ge-si alloys up to 1300 degrees k. *J. Appl. Phys.*, 35(10):2899, 1964.

28 ICT-Energy - Nanoscale Energy Management Concepts

to expand over the next decades.

**Author details**

Douglas Paul

**References**

times. These systems are aimed at solar farms which in sunny regions of the earth have the potential to produce power stations at the 100s of MW scale. Such concentration, however, results in the photovoltaic cells being heating to very high temperatures that results in large thermal cycling which ultimately produces failures reducing the lifetime of the systems. By integrating a thermoelectric with the photovoltaic, not only can electricity be generated from the heat thereby providing a joint and therefore improved system efficiency closer to the Carnot limit but more importantly, the thermoelectric cools the photovoltaic reducing the thermal cycling extremes and increases the lifetime of the system. This increase in lifetime results in substantially cheaper cost per Watt which is the major driver for photovoltaics. It

At present thermoelectrics requires a "killer application" before volume manufacture will result in widespread use. The automotive applications at present appear to be the major application driver, more by legislation rather than market but it is clear that the potential requirements for thermoelectric generators will improve as fossil fuel prices increase in the future. The autonomous sensor market may well also drive thermoelectrics but the real problem that must be solved is to find sustainable thermoelectric materials. Tellurium is the 9*th* rarest element on earth predicted to run out before 2020 with the present use and so a key requirement for future thermoelectrics is that Te-free materials can be found that can be cheaply produced and with a high ZT and power factor. There is presently an enormous amount of research in thermoelectrics especially aiming to find Te-free materials and to produce modules for higher temperature applications such as the automotive energy harvesting. For students starting in a research career, this is one research area that is likely

is clear this will be a developing application for thermoelectrics in the future.

University of Glasgow, School of Engineering, Rankine Building, U.K.

[1] J.M. Ziman. *Electrons and Phonons*. Qxford University Press, 1960.

*Phys. Rev.*, 181(3):1336–1340, May 1969.

113(4):1046 – 1051, 1959.

Springer, 1980.

7(2):105–114, 2008.

[2] Melvin Cutler and N. F. Mott. Observation of anderson localization in an electron gas.

[3] J. Callaway. Model for lattice thermal conductivity at low temperatures. *Phys. Rev.*,

[4] B.R. Nag. *Electron Transport in Compound Semiconductors*, volume 11 of *Solid State Sciences*.

[5] G. J. Snyder and E. S. Toberer. Complex thermoelectric materials. *Nature Materials*,

