Thermoelectric Materials

Part of the Springer Handbooks book series (SPRINGERHAND)

Abstract

Thermoelectricity is one of the oldest phenomena to be observed in semiconductors, with discovery of the various thermoelectric effects dating back to the early part of the 19th century. These effects manifest themselves as the appearance of a voltage in a circuit comprised of two different conductors due to a temperature difference (Seebeck effect ), or as the absorption and evolution of heat at the junction of two different materials under electrical current excitation (Peltier effect ). These effects can be utilized in devices to generate electrical power from waste heat or to provide solid state cooling, respectively.

This chapter reviews the main factors governing thermoelectric effects in solids, and how these factors may be manipulated to produce materials with high thermoelectric figure of merit. The first portion of the chapter covers the main features that determine electrical and thermal transport in crystalline semiconductors, while the latter portion discusses several new approaches to this old problem that hold promise for highly efficient thermoelectric materials in the future.

Thermoelectricity encompasses the collective effects that involve the conversion of a temperature difference to electricity and vice versa. The thermoelectric process is fundamentally a microscopic one involving the transport and exchange of energy by and between electrons and lattice vibrations, or phonons, in a solid. This process can be described in terms of the Seebeck, Peltier, and Thomson coefficients. As such these effects may be implemented to generate electrical power from primary or waste sources of heat, or inversely to provide all-solid-state heat pumping, i. e., climate control, under electrical activation. The extent to which thermoelectricity might impact our utilization of energy in the United States is provided by an examination of the flow and usage of energy in the transportation, industrial, commercial, and residential sectors in the United States, as provided by Lawrence Livermore Laboratory, United States Department of Energy [57.1]. These data indicate that almost two-thirds of the energy produced is rejected mainly in the form of waste heat. This is not really surprising to us, as we all learn at an early age to keep well away from the exhaust pipe of even an idling automobile and to never open the cap on the radiator of a vehicle whose engine is running; and we all have witnessed the great clouds of bellowing steam emerging from the stack of a nuclear- or coal-burning power plant. This rejected energy represents a tremendous opportunity – in some sense, it is like a new energy source, prodigious in size, available almost for free and waiting to be used. To appreciate the size of this energy source, the energy content available in this pathway is equivalent to that in over 400 billion gallons of gasoline [57.2]. It is here, in the recovery of wasted thermal energy, where thermoelectricity can play a pivotal role. Conversion of this heat to electricity, even at modest fractions, can improve significantly the overall efficiency of energy utilization in the US and worldwide, and thereby both reduce dependence on foreign sources of fossil fuels and significantly decrease carbon emissions. It is principally for this reason that the field of thermoelectricity, which has always been of interest scientifically if not technologically, has grown substantially in the last ten years.

57.1 Overview of the Field

Thermoelectric energy conversion systems use the thermoelectric effect to convert heat, either from a primary source such as a burner, or waste heat, such as that in the exhaust stream of an automobile or power plant, to useful electricity. The efficiency for conversion of heat to electricity is determined by the dimensionless figure of merit \(zT=\alpha^{2}\sigma T/\kappa\), where α is the Seebeck coefficient , σ the electrical conductivity, κ the thermal conductivity of the materials comprising the thermoelectric device, and T the absolute temperature. For a thermoelectric device operating between a heat source at temperature TH and a heat sink at temperature TC, the efficiency is given by [57.3]
$$\varepsilon=\frac{\Updelta T}{T_{\mathrm{H}}}\left(\frac{\sqrt{1+zT_{\text{ave}}}-1}{\sqrt{1+zT_{\text{ave}}}+\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}}}\right),$$
(57.1)
where \(T_{\text{ave}}=(T_{\mathrm{H}}+T_{\mathrm{C}})/2\). Figure 57.1 is a surface plot of this relationship for \(T_{\mathrm{H}}={\mathrm{700}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\) (973 K) as a function of TC and zT. We see that a conversion efficiency exceeding 25% under these conditions can be achieved over a wide range of figures of merit and cold-side temperatures.
Fig. 57.1

Efficiency of a thermoelectric generator with a hot-side temperature of 700C (973 K) as a function of cold-side temperature and figure of merit zT

Thermoelectric materials research dates back to the time of Seebeck himself, who studied the effect that bears his name in a large number of metals and minerals [57.4]. Subsequent studies were carried out by Haken [57.5] and Altenkirch [57.6] in Germany in the early part of the 20th century. The modern era of thermoelectricity dates back to the work of Ioffe (Institute of Semiconductors, USSR) [57.7] and Goldsmid (General Electric Research Laboratories, Great Britain) [57.3]. Through combined experimental and theoretical work it was well established by 1970 that the best thermoelectric materials, such as PbTe and Bi2Te3 with zT ≈ 0.5–1.0, were narrow gap semiconductors with large charge carrier effective masses and comprised of heavy-atom elements arranged in high-symmetry crystal structures.

In spite of a plethora of studies, both theoretical and experimental, on new thermoelectric materials, for the next quarter of a century the field remained somewhat quiescent, with no substantial improvements in the figure of merit. It was recognized that the thermoelectric problem was one in which the figure of merit requires a combination of contraindicated physical properties: a set of properties which nature is not disposed to provide in a single material, for instance high electrical conductivity and large Seebeck coefficient, or high electrical conductivity and low thermal conductivity.

The situation changed rapidly in the mid-1990s for a number of different reasons. Firstly, due to a greater awareness of the insecurity of our energy future and the impact of energy usage on the precarious state of the environment, the scientific and technical communities began to focus more heavily on the science and technology of energy processes and energy-related materials; secondly, the field began to benefit from the use of more sophisticated synthesis techniques and sensitive characterization methods, especially at the nanoscale; and finally, and perhaps most importantly, the scientific community developed an increased understanding, at a fundamental level, of the nature of electrical and thermal transport processes in solids, two examples being the theoretical treatment of Hicks and Dresselhaus [57.8, 57.9] of thermoelectricity in quantum-confined systems and the origination by Slack [57.10] of the concept of phonon-glass-electron-crystal .

From (57.1), it can been seen that commonly available thermoelectric materials like Bi2Te3 and PbTe, with zT values on the order of unity or less, have a conversion efficiency of under 10%. Clearly the most direct means of increasing the efficiency is to improve upon the transport properties that comprise the figure of merit. We now look at this relation in somewhat greater detail
$$z=\frac{\alpha^{2}\sigma}{\kappa}=\frac{P}{\kappa}\;.$$
(57.2)
Here κ is the total thermal conductivity and consists of the sum of the electronic (κe) and lattice, or phonon (κp) contributions. The numerator term, the electrical part of the figure of merit, is called the power factor (P).

One of the oldest [57.7] and most effective ways of increasing z is by alloying to form solid solutions such as \(\mathrm{Pb_{1-{\mathit{x}}}Sn_{\mathit{x}}Te_{1-{\mathit{y}}}Se_{\mathit{y}}}\) or \(\mathrm{Bi_{2(1-{\mathit{x}})}Sb_{2(1-{\mathit{x}})}Te_{3(1-{\mathit{y}})}Se_{3{\mathit{y}}}}\). The basis for this approach is the strong phonon scattering that occurs due to the mass difference on one or both of the binary compound lattice sites. In this approach the long-range periodicity of the lattice is preserved, but there are intense short-range perturbations. Electrons, with wavelengths many times the interatomic distance, are essentially uninfluenced by these perturbations, while phonons, the characteristic wavelength of which is on the order of the interatomic distance, can be strongly scattered by these local distortions in atom mass and size. The net result is that the ratio σ ∕ κp, or more particularly μ ∕ κp, where μ is the carrier mobility, is increased substantially, leading to an enhancement of z.

Within the last decade there have been many novel approaches to designing new thermoelectric materials with values of figure of merit beyond those possible using the solid-solution method. These approaches can be broadly grouped into two categories: 1. those that seek to increase the Seebeck coefficient α; and 2. those that seek to increase the ratio of σ ∕ κ. An example of the former includes the use of low-dimensional systems [57.11, 57.8, 57.9] in which the altered density of electron states leads to enhanced α; examples of the latter include phonon-glass electron-crystal structures such as filled skutterudite [57.12] and clathrate [57.13] compounds with glass-like lattice thermal conductivity, and superlattice structures [57.14] with increased phonon scattering by interfaces.

The main challenge in any methodology seeking to reduce the lattice thermal conductivity is maintaining adequate carrier mobility, and hence electrical conductivity. Magnetic scattering, for instance, using rare earth ions or transition metal ions with partially filled d-shells, can dramatically reduce the lattice conductivity, but is inevitably accompanied by a strong reduction in carrier mobility [57.15]. Even the filled skutterudites, with their notable reduction in κp, suffer a strong reduction in mobility upon filling and reach high z only by an alteration of band structure that causes an increase in Seebeck coefficient relative to their unfilled counterparts.

57.2 Semiconductors as Thermoelectric Materials

57.2.1 Electronic Properties of Charge Carriers in Semiconductors

An intrinsic semiconductor is defined as a material in which, at T = 0, completely filled electron energy levels (known collectively as the valence band) are separated in energy from energy levels that are empty (known collectively as the conduction band); thus, at zero temperature, a semiconductor is insulating, because there are no empty states nearby in energy into which an electron may move under application of an electric field. As temperature is increased, electrons may be thermally excited across the energy gap to occupy states in the conduction band. Electrons so promoted in energy are then free to move to other nearby states in the conduction band, and thus can contribute to electrical conductivity. At the same time, the empty states left behind in the valence band, called holes (equal in number to electrons promoted), can also contribute to conduction as a positive charge carrier. The two types of charge carriers in an intrinsic semiconductor, electrons (n) and holes (p), move through the lattice with distinct mobilities, μe and μh respectively, and thus in an intrinsic semiconductor, the total electrical conductivity, σ, is given by the sum of the partial conductivities of electrons (σe) and holes (σh)
$$\sigma=\sigma_{\mathrm{e}}+\sigma_{\mathrm{h}}=ne\mu_{\mathrm{e}}+pe\mu_{\mathrm{h}}\;.$$
(57.3)
Because electrons and holes are fermions, the probability of their occupation of a given energy state is determined by Fermi–Dirac statistics. The Seebeck coefficient of an intrinsic semiconductor, on the other hand, is determined by the partial Seebeck coefficient of the electrons and holes, weighted by their conductivities, according to the relation
$$\alpha=\frac{\alpha_{\mathrm{e}}\sigma_{\mathrm{e}}+\alpha_{\mathrm{h}}\sigma_{\mathrm{h}}}{\sigma_{\mathrm{e}}+\sigma_{\mathrm{h}}}\;.$$
(57.4)
In an intrinsic semiconductor, since by definition n = p, this relation simplifies to one in which the partial Seebeck coefficients are weighted by the carrier mobilities
$$\alpha=\frac{\alpha_{\mathrm{e}}\mu_{\mathrm{e}}+\alpha_{\mathrm{h}}\mu_{\mathrm{h}}}{\mu_{\mathrm{e}}+\mu_{\mathrm{h}}}\;.$$
(57.5)
Once again, the partial Seebeck coefficients of electrons and holes are, as we shall see in a moment, determined by Fermi–Dirac statistics. Here we note, however, that the sign of the Seebeck coefficient is the same as that of the sign of the charge carrier, that is, negative for electrons and positive for holes. Thus in an intrinsic semiconductor, the total Seebeck coefficient is always less than the partial Seebeck coefficient of either the electrons or the holes. For this reason, virtually all useful thermoelectrics are designed using extrinsic doping to produce n- and p-type materials respectively.
Going a step further, the mobility of charge carriers may be written as
$$\mu=\frac{e\tau}{m^{\ast}}\;.$$
(57.6)
Here τ is the scattering time of the charge carriers and m is the carrier effective mass. We make a further assumption that the charge carrier scattering time depends on the carrier energy E as
$$\tau(E)=\tau_{\mathrm{o}}E^{\lambda}\;.$$
(57.7)
Returning now to the equation for the figure of merit, using the definition of the conductivity this equation may now be written, for an n-type material
$$z=\frac{\alpha^{2}\sigma}{\kappa}=\frac{\alpha^{2}(ne\mu)}{\kappa}=(\alpha^{2}ne)\frac{\mu}{\kappa}\;.$$
(57.8)
In this form, z depends on one quantity, α2ne, which is determined by the electronic band structure, and a second quantity μ ∕ κ, which depends on the nature of electron and phonon scattering. To analyze the figure of merit more deeply, we break it into these two pieces and discuss each in turn.
Fermi–Dirac statistics describe the probability of occupation of electron (and hole) states in semiconductors. At temperature T > 0, there is finite probability that an electron can occupy the conduction band or a hole can occupy the valence band. In this situation, the electron concentration and Seebeck coefficient are given by the expressions [57.3]
$$\begin{aligned}\displaystyle n&\displaystyle=\frac{4}{\sqrt{\uppi}}\left(\frac{2\uppi m^{\ast}k_{\mathrm{B}}T}{h^{2}}\right)^{\frac{3}{2}}F_{\frac{1}{2}}\\ \displaystyle\alpha&\displaystyle=\mp\frac{k_{\mathrm{B}}}{e}\left[\frac{\left(\frac{5}{2}+\lambda\right)F_{\frac{3}{4}+\lambda}}{\left(\frac{3}{2}+\lambda\right)F_{\frac{1}{2}+\lambda}}-\eta\right].\end{aligned}$$
(57.9)
Here the Fs are the so-called Fermi integrals of order r
$$\begin{aligned}\displaystyle F_{\mathrm{r}}&\displaystyle=\int_{0}^{\infty}x_{\mathrm{r}}f(\eta)\mathrm{d}x\quad\text{ where}\\ \displaystyle x&\displaystyle=\frac{E}{k_{\mathrm{B}}T}\quad\text{and}\quad\eta=\frac{E_{\mathrm{f}}}{k_{\mathrm{B}}T}\;.\end{aligned}$$
(57.10)
The Fermi–Dirac distribution f is given by
$$f=\frac{1}{1+\exp(x-\eta)}\;.$$
(57.11)
As usual h and kB designate the Planck and Boltzmann constants respectively, and Ef is the charge carrier Fermi energy. The reduced Fermi energy η is the Fermi energy as measured from the edge of the band, i. e., η = 0 at the bottom of the conduction for electrons and is negative for Fermi energy in the gap, and similarly η = 0 for holes at the top of the valence band and is negative again for Fermi energy in the gap. The Fermi energy, and thus η, takes on positive values as it moves either into the conduction band for electrons or into the valence band for holes. Using these equations, for a given electron scattering coefficient λ and carrier effective mass m, one can calculate the Seebeck coefficient α and the quantity α2n that enters directly into the thermoelectric figure of merit.
Figure 57.2 shows the results of such calculations and compares them to the results obtained using classical statistics, i. e., treating the electrons in a solid as if they were like molecules in a gas. Some interesting and important differences emerge. For \(\eta<-2\), i. e., for Fermi energy well in the gap, the classical and Fermi–Dirac statistics give identical results. This is the nondegenerate case of a lightly doped semiconductor. For classical statistics the Seebeck coefficient falls more rapidly and actually reaches zero at η = 2. For Fermi–Dirac statistics, the Seebeck coefficient is diminished as the Fermi level moves into the conduction or valence band, but it never reaches zero. Similarly, the important quantity α2n is maximized for Fermi energy right at the band edge for classical statistics, whereas for the more general Fermi–Dirac case, the maximum occurs for Fermi energy one unit of kBT into the band.
Fig. 57.2

Seebeck coefficient (a) α and (b) α2n as a function of reduced Fermi energy for holes with effective mass \(m^{\ast}={\mathrm{0.3}}\)

Of course, experimentally, one does not directly control the Fermi energy; rather, one controls the carrier concentration through doping. To see this more clearly, Fig. 57.3 plots the quantity α2n versus carrier concentration n, for Fermi–Dirac statistics. Here we clearly see the origin in the peak in the power factor α2σ, predominantly arising due to the inverse relationship between α and n.
Fig. 57.3

The quantity α2ne as a function of carrier concentration at 300 K for a carrier effective mass \(m^{\ast}={\mathrm{0.3}}\)

Thus Fermi–Dirac statistics give a neat interpretation of the quantity α2n and in particular what value should be chosen for the Fermi energy and can be used as a guide to approximate what level of doping is required in order to optimize the thermoelectric figure of merit. In practice, because others factors such as the dependence of effective mass on doping level and mixed carrier scattering mechanisms, usually the optimum doping level for maximum power factor is determined by trial and error.

57.2.2 Semiconductors with Low Intrinsic Thermal Conductivity

While a reasonably clear understanding of the thermoelectric power factor and the elements that influence it can be obtained with knowledge of the band structure of a material, the dominant carrier scattering mechanism, and the use of carrier statistics, that of course is only part of the thermoelectric problem – we have not yet dealt with the denominator of the figure of merit; the thermal conductivity. While to first order there is much to be learned from decoupling the treatment of the power factor from that of the thermal conductivity, in reality all of the transport properties comprising the figure of merit are intertwined and interdependent at some level.

Serious study of the thermal conductivity of crystalline solids was begun before World War II, including the work of Akhiezer [57.16], Leibfried and Schloemann [57.17], and Lawson [57.18]. A basic conclusion of all of these studies was that the lattice thermal conductivity of crystals near and above room temperature obeyed the general relation [57.19]
$$\kappa_{\mathrm{L}}\propto\frac{\theta^{3}}{\gamma^{2}T}\;,$$
(57.12)
where θ is the Debye temperature , γ the Grüneisen constant , and T the absolute temperature. On the basis of this simple relation, one can thus predict that materials with high Debye temperature and/or small Grüneisen parameter will have high thermal conductivity, and vice versa. Since the Debye temperature depends inversely on the average atom mass, this further implies that low thermal conductivity will be favored in materials comprised of heavy mass atoms. Subsequently, Slack [57.20] extended this relationship to include a dependence of crystal complexity, i. e., number of atoms in the unit cell, and showed that crystal structures with large unit cells generally possess lower thermal conductivity due to a more complex phonon spectrum. Taken together, this information suggests that the place to look for good thermoelectric materials is amongst semiconductors with low Debye temperature, large numbers of atoms in the unit cell, heavy atom masses, and large Grüneisen coefficients. At the same time, of course, these compounds must possess high thermoelectric power factor, as determined by the equations outlined in Sect. 57.2.1 above.
In practice, there are thousands of possible compounds and dozens of different crystal structure types that must be considered for possible choices for thermoelectric materials, but the power of classifying materials according to their crystal structure and bonding can be illustrated by comparing some results for diamond/zincblende and rock salt structure compounds. It may be noted that the same properties that favor low thermal conductivity (heavy atom masses and low Debye temperature) are also typical of lower melting point compounds, and indeed (Fig. 57.4) there is a strong correlation between lattice thermal conductivity and melting point, at least within families of compounds such as the diamond/zincblende structure and the rock salt structure. The magnitude of the conductivity, however, does depend on the nature of bonding between atoms, as clearly the covalently bonded diamond/zincblende structure compounds and elements have higher thermal conductivity than the ionically bonded rock salt structures. On this basis alone one would naturally be led to choose the rock salt structure compounds over the zincblendes due to the lower thermal conductivity. On the other hand, as mentioned above, we must ultimately find materials that not only have low lattice thermal conductivity, but also high charge carrier mobility, in order that the electrical conductivity may remain high, and this tends to be favored in covalently bonded crystals. It is the hallmark of research on thermoelectric materials that one is constantly confronted, in different forms and guises, an inherent contraindication in properties that can frustrate the search for high z.
Fig. 57.4

Lattice thermal conductivity versus melting point for a host of diamond, zincblende and rock salt structure compounds

57.2.3 Semiconductor Solid Solutions

While the above approach for finding compounds with low intrinsic thermal conductivity was very successful, it resulted in very few materials being identified with high z. Researchers quickly noted that z could be increased further, however, by lowering the thermal conductivity using defects or impurities, as long as the electrical properties could be largely maintained. By far the most successful method of achieving this result was by realizing isoelectronic solid solutions of elemental and compound semiconductors. The use of solid solutions to reduce thermal conductivity is rooted in the concept that impurities, when doped into a host lattice, will scatter phonons due to differences in their mass and size with respect to the host atoms. Such behavior was predicted originally by Pomeranchuk [57.21], was explored theoretically first by Klemens [57.22], and was experimentally demonstrated on a host of materials systems. In the case of thermoelectrics, as we have discussed, it is imperative that when such impurities are introduced into the lattice, they disturb the transport of electrons as little as possible. For example, dopants introduced into a semiconductor to increase the charge carrier concentration can at high concentrations reduce charge carrier mobility due to ionized impurity scattering. Solute atoms that have the same valence electron count as the host atoms in a lattice, on the other hand, are neutral impurities and interact with charge carriers much less strongly; the resulting reduction in mobility can be quite small. By choosing atoms in the same column of the periodic table (but obviously from a different row), reduction in lattice thermal conductivity can be achieved while largely maintaining the good electrical properties. Moreover, because of the same valence electron count, these atoms can be substituted at large fractions, and in many cases will form a complete solid solution.

The isoelectronic substitution approach was applied very successfully to both PbTe [57.7] (PbTe1−xSe x and PbTe1−xS x ) and Bi2Te3 [57.8] (Bi2Te3−xSe x , Bi2Te3−xS x , and Bi2−xSb x Te3), and was also successfully implemented in the group IV semiconductor system Si-Ge [57.23]. Several theoretical models have been developed to account for the lattice thermal conductivity of alloy solid solutions. Figure 57.5 shows an example of the experimental data for PbTe1−xSe x in comparison to the model predictions of Abeles [57.24]. As expected, the reduction in thermal conductivity, over a factor of two in this case, is maximum near 50% substitution, when the mass disorder in the lattice is at its largest.
Fig. 57.5

Lattice thermal conductivity of the PbTe1−xSe x solid solution. The reduction in thermal conductivity is maximized near 50% solute concentration, the point at which mass disorder in the lattice is largest

57.3 New Concepts in Thermoelectric Materials Design

By the mid-20th century, most of the ideas discussed above on how to optimize the thermoelectric properties of materials were well known and had been successfully implemented to identify PbTe and Bi2Te3 solid solutions as good thermoelectric materials. As part of the electronics revolution, thermoelectricity seemed to be on the cusp of explosive discovery and development as a new technology for generating electrical power and providing solid-state heating and cooling.

It seemed that researchers simply had to apply the new concepts and knowledge to find materials even better than the lead and bismuth tellurides. In spite of that promise, however, very little progress was made over the next several decades, and even by 1990 essentially no new thermoelectric materials with properties superior to the heritage materials were discovered. On top of this, the very factors that make these heritage materials such good thermoelectrics, for example heavy masses and weak atomic bonding, also mean that they tend to be mechanically poor, and comprised of elements that are not in great abundance in the Earth’s crust. Thus, even these good thermoelectric materials found limited applications and thermoelectricity was a field on the verge of becoming relegated to the backwaters of scientific inquiry.

By the late 20th century, however, a number of factors and influences came together to lead to a refocusing of efforts on discovery of new thermoelectric materials. While indeed over the past several decades few if any new thermoelectric materials of note were discovered, steady progress was made at the level of advancing our fundamental understanding of thermal and electronic transport in solids. At the same time, new ideas, such as quantum confinement and nanoscience, were introduced and explored in a variety of materials systems, and new methods of synthesis of materials were perfected. Thus several tools and fresh approaches could be brought to bear on the field – one just needed the impetus for doing so. That impetus was provided by a new sensitivity toward the utilization of energy and its potential impact on the climate and environment of the Earth. Suddenly, this old idea of converting heat to electricity, mostly viewed upon as a laboratory curiosity, became a new way to generate power, increase efficiency, and decrease emissions of potentially harmful pollutants. When large sources of federal funding became available for research, scientists were quick to bring to bear a host of these new ideas and new approaches to the field of thermoelectricity. Here, in the interest of space and to keep the discussion focused, we consider three new concepts that have emerged over the last two decades and that continue to drive thermoelectric materials research today, and we describe each of these new approaches in terms of model materials systems:
  • Phonon-glass-electron crystal (PGEC )

  • Bulk nanostructured materials

  • Crystals with large anharmonicity.

57.3.1 Phonon-Glass Electron Crystal: Skutterudites

While the solid-solution approach can reduce lattice thermal conductivity strongly, the magnitude of the thermal conductivity typically is still several times larger than that observed in materials with very low thermal conductivity, such as glasses and other amorphous structures. Study of the thermal conductivity of the latter materials has its own long and storied history, dating back to at least the studies of Zeller and Pohl [57.25], who noted the vast difference in thermal conductivity between crystalline and amorphous SiO2. A significant amount of work was subsequently performed on such materials that described heat transport by a phonon-tunneling-type mechanism. Eventually, the concept of minimal thermal conductivity (MTC) was introduced that described how in such amorphous and highly disordered solids thermal conductivity reaches a theoretically minimum value corresponding to a phonon mean free path on the order of the lattice spacing [57.26].

While the thermal transport properties of glassy systems is an interesting topic in its own right, it did not seem to bear importance relative to the thermoelectric problem; most amorphous solids are poor electrical conductors and are not good candidates for thermoelectrics. Slack, however, noted that there were a few crystalline solids that exhibited glass-like thermal conductivity – a notable example being YB66. This compound displays galleries of boron atoms, between which Y ions can tunnel. In spite of its crystalline character, the thermal conductivity reaches theoretically minimum values. Slack pointed out that perhaps crystalline semiconductors may exist that have not only good thermoelectric properties, but at the same time also exhibit MTC behavior, and suggested that skutterudite compounds could be such a material.

Like many thermoelectric materials, skutterudites were first studied in the Soviet Union by Dudkin et al. [57.27, 57.28, 57.29, 57.30]. They found that this family of compounds, typified by the composition CoSb3, could be doped both n- and p-type and displayed a good combination of high mobility, large Seebeck coefficient, and large thermoelectric power factor. However, in spite of the rather complex crystal structure (Fig. 57.6) consisting of a cubic lattice Co atoms containing Sb four-membered square rings in six of the eight quadrants of the unit cell, the lattice thermal conductivity of skutterudites was found to be too large to yield high z. This early work did not explore in detail any mechanisms to reduce the thermal conductivity.
Fig. 57.6a,b

The skutterudite crystal structure. A prototypical compound is CoSb3; Co atoms are brown and Sb are gray. The Sb atoms reside in square planar rings occupying six of the eight quadrants in the unit cell. In the unfilled compound the remaining two quadrants are empty (a); in the filled version they are occupied by a rare earth or alkaline earth atom (b)

More than a decade later, Jeitschko [57.31] showed the skutterudite structure could be filled by placing lanthanide ions in the two vacant quadrants in the unit cell, and that these ions exhibited large x-ray displacement parameters, indicating that they were loosely bound or rattled about their ionic positions. Some 15 years later, with tremendous insight, Slack suggested that these filled skutterudites might exhibit minimal thermal conductivity due to the rattling phonon mode of the lanthanide ion. A strong reduction in lattice thermal conductivity of the filled skutterudite CeFe4Sb12 relative to its unfilled parent compound, CoSb3, was experimentally demonstrated by Morelli and Meisner [57.12] shortly thereafter. Skutterudites were the subject of intense scrutiny, both from the point of view of their interesting physics and chemistry, and for their potential thermoelectric applications, and soon it was shown that these compounds could exhibit a figure of merit in excess of unity at high temperature.

Crucial to the performance of skutterudites is the combination of low lattice thermal conductivity in the presence of good semiconducting electrical properties. Because the filling ion (which in addition to an element from the lanthanide series, may also be an alkaline earth ion) donates electrons to the conduction band, its presence can also modify the electrical properties. By simultaneously substituting Fe for Co, the maximum filling fraction can be adjusted and the electrical properties controlled to produce either n- or p-type materials of desired carrier concentration [57.32]. Recognizing that the rattling behavior exhibited by the filler atom was a form of resonant scattering of phonons with a characteristic frequency determined by the mass of the filler atom and the size of the cage, researchers used combinations of fillers (double or even triple filling) to provide different characteristic resonant frequencies, thus producing the ability to scatter wider ranges of the phonon spectrum and reduce the thermal conductivity even further [57.33, 57.34].

Research and development on skutterudites has continued up to the present day, and impressive zT values in excess of 1.5 for n-type materials and in excess of unity for p-type materials have been achieved. Prototype generators based on skutterudites for exhaust gas-driven power generation on vehicles have been built.

57.3.2 Bulk Nanostructured Thermoelectrics: Pb-Chalcogenide Nanocomposites

About the same time that Slack introduced the PGEC concept, researchers began to contemplate the role that quantum confinement, low-dimensionality, and nanostructuring may play in thermoelectricity. The papers by Hicks and Dresselhaus [57.8, 57.9] considered the influence of the modification of the electron density of states that occurs in one-dimensionally (quantum wire) and two-dimensionally (quantum well) confined systems, and these theoretical studies launched intense investigation of these topics. While the promising early results of Harman et al. [57.35] and Venkatasubramanian et al. [57.14] have not been reproduced, the subsequent studies on nanostructured materials that grew out of these efforts were to bear fruit in the coming years.

An illustration of the positive effects of nanostructuring is provided by the work on so-called LAST materials and hierarchically architectured nanocomposite thermoelectrics. The former is an acronym for compounds based on the PbTe-AgSbTe2 (lead–antimony-silver-telluride) family and was extensively studied by Kanatzidis et al. [57.36]. The unusual thermoelectric properties of these materials were noted much earlier by both Fleischmann [57.37] and Irie et al. [57.38]. As AgSbTe2 is isostructural with PbTe (face-centered cubic rock salt structure) the PbTe-AgSbTe2 compounds were originally thought to form a complete solid solution. The work of the Kanatzidis group showed, however, that small Ag-Sb rich nanoclusters form in these materials, and in fact this morphology is responsible for an enhancement of the thermoelectric properties, primarily through a reduction in thermal conductivity by scattering of phonons at the interface between the host matrix and the secondary phase. Phonon scattering by inclusions on the nanoscale has a long history of study, dating back to at least the work of Slack on CaCl2 precipitates in KCl [57.39], and in fact the use of nanoscale inclusions to reduce the thermal conductivity of Si-Ge thermoelectric alloys was studied as early as 1993 [57.40]. The nanostructured LAST compounds exhibited a thermoelectric figure of merit of zT = 1.6.

While nanoscale inclusions can be effective scatterers of medium-wavelength phonons in solids, phonons with frequencies both higher and lower than those that interact strongly with inclusions can still transport heat. On this basis, workers began conceiving of materials systems that could scatter a broader range of phonon frequencies, and thus reduce the thermal conductivity further. It was in this spirit that the concept of hierarchical structuring of defects in thermoelectric materials was born (Fig. 57.7). Here the idea is that by combining different types of defects/impurities, each type of which scatter phonons in different frequency ranges, almost all heat transport by phonons can be blocked, just as different optical filters can block different wavelengths of light and when combined can completely block photon transmission. This concept was realized experimentally in the work of Biswas et al. [57.41] in which atomic-scale defects, nanostructured second phases, and grain boundaries were combined to scatter high-, medium-, and long-wavelength phonons respectively. The base materials system used to achieve this was once again PbTe, into which was doped SrTe, another rock salt structure compound. The solubility limit of Sr for Pb in PbTe, however, is rather small, on the order of 1%. At first some Sr substitutes for Pb (atomic scale defects), but as the concentration is increased, SrTe nanoclusters form in the material (nanoscale defects). By processing these materials using spark plasma sintering, a fine grain size can be maintained (mesoscale defects). The resulting hierarchy in defect size reduces the thermal conductivity in these materials down to close to minimal thermal conductivity values. As a result, the thermoelectric figure of merit of optimally doped samples reaches values as high as 2.2.
Fig. 57.7

Hierarchical structuring of thermoelectric materials. Defects and impurities on different length scales scatter phonons of different wavelengths. Grain boundaries on the mesoscale (left; scale bar ≈ 103–104 nm) scatter long-wavelength phonons; nanoscale precipitated second phases (center; scale bar ≈ 101–103 nm) scatter medium-wavelength phonons; and atomic scale impurities (right; scale bar ≈ 100–101 nm) scatter short-wavelength phonons

57.3.3 Crystals with Large Anharmonicity: Tetrahedrites

While impressive results have been obtained using nanostructuring, there are some potential drawbacks with this approach. The composition, distribution of second phases, and grain sizes must be carefully controlled in order to achieve the maximum reduction in thermal conductivity. The processing of these materials is, while straightforward, time consuming. Finally, the stability of these materials with thermal and mechanical cycling is a still-open question.

For these reasons, workers have been looking for other approaches to finding solids with low thermal conductivity. One such approach is to not use any defects or impurities at all – instead, to try to find materials with built-in, intrinsically low thermal conductivity. Of course, as discussed earlier, many of the main drivers of low thermal conductivity, such as complex crystal structures and heavy atom masses, have been known for many decades. Somewhat overlooked, however, has been the nature of phonon scattering, and how it might be maximized intrinsically based on crystal structure and bonding in a solid.

In a simple model of a crystal lattice, atoms are bonded together like masses connected by springs. In a quantum-mechanical picture of the lattice, the vibrations of atoms are quantized and are collective modes known as phonons. For a perfectly harmonic lattice, i. e., one in which the potential energy of each ion is a quadratic function of its displacement, there are no interactions between phonons, and the intrinsic thermal resistance is zero. Lattice thermal resistivity in solids arises due to anharmonicity in the lattice vibrational spectrum – i. e., vibrations for which the potential energy has additional terms of higher order than quadratic. For instance, a cubic term in the potential energy gives rise to anharmonic three-phonon processes, two examples of which are given in Fig. 57.8. Four-phonon and higher-order processes will arise from terms quartic and higher.
Fig. 57.8a,b

Two types of three-phonon anharmonic processes that give rise to intrinsic lattice thermal resistivity in solid. (a) A phonon of wavevector q1 is destroyed and two phonons of wavevectors q2 and q3 are created (phonon creation); (b) two phonons of wavevector q1 and q2 are destroyed and a phonon of wavevector q3 is created (phonon annihilation)

A measure of the anharmonicity in a crystal lattice is given by the Grüneisen coefficient γ. The ideally harmonic crystal would have γ = 0 for all phonon modes and frequencies; this is an ideality that does not exist in nature. The larger the value of γ, the stronger the anharmonicity in the lattice. Thus, a good place to look for crystalline solids with low intrinsic thermal conductivity is in those materials with highly anharmonic phonon spectra.

But what gives rise to anharmonicity ? This very fundamental question has been of intense interest to researchers in recent years. One very interesting and fruitful approach is the concept of the effect of lone pair electrons on vibrational modes in solids. The seed of this idea goes back to the work of Zhuze [57.42] in the Soviet Union, who noticed the large difference in thermal conductivity between certain compounds depending on their composition and the valence state of the atoms comprising them, and in particular those containing Sb ions. The electronic configuration of Sb is [Kr]4d105s25p3. In certain coordinations of Sb in the lattice, all five of the valence electrons participate in bonding; an example is the zincblende-like compounds Cu3SbSe4, in which each Sb atom is tetrahedrally coordinated. On the other hand, in other compounds, for instance Cu3SbSe3, the Sb is trigonally coordinated, and only three of the valence electrons participate in bonding; the remaining two electrons reside close to the Sb ion in an unbonded configuration. This configuration of unbonded electrons around Sb (or indeed any other ion) is known as the lone pair. Zhuze suggested that the electron clouds of overlapping Sb lone pair electrons gives rise to nonlinear forces in the lattice vibrations, i. e., give rise to anharmonicity. This effect was recently invoked by to explain the low intrinsic thermal conductivity of AgSbTe2 [57.43]. Skoug and Morelli [57.44] showed that the thermal conductivity of large families of compounds exhibiting pnicogen atom lone pairs could be understood in this context, and were able to relate the degree of thermal resistivity with the distance of the lone pair electrons from the nucleus of the atom, with those configurations in which the lone pair was furthest from the nucleus having the strongest reduction in thermal conductivity. Ozolins et al. [57.45] formalized the connection between the presence of the lone pair and the degree of anharmonicity using calculations of the lattice dynamics for several compounds.

A very interesting example of how this effect can manifest itself in enhanced thermoelectric behavior is provided by the family of compounds known as tetrahedrites. The base composition of tetrahedrite is Cu12Sb4S13. Its crystal structure is cubic with 29 atoms in the unit cell. Tetrahedrite possesses several key structural features that have strong implications for its thermoelectric properties. First, the Sb atoms are trigonally coordinated and do indeed display the classic lone pair as described above. Additionally, Cu occupies two distinct crystallographic sites in the lattice; one in which the Cu atom is tetrahedrally coordinated by sulfur, and a second site in which the Cu ion resides in a trigonal plane of sulfur atoms; six Cu atoms are supposed to reside in each site in the lattice. Guided by electronic band structure calculations, Lu et al. [57.46] showed that while the base composition Cu12Sb4S13 is metallic, the substitution of divalent Zn for Cu can yield semiconducting behavior. At the same time, the lattice thermal conductivity exhibited values below \({\mathrm{0.5}}\,{\mathrm{Wm^{-1}K^{-1}}}\), i. e., close to the minimal thermal conductivity of a solid. By optimizing the Zn concentration they demonstrated that tetrahedrites can reach zT values close to 0.9. Since then, several other studies [57.47, 57.48, 57.49] investigating different atoms substitutions and different methods of processing these materials has resulted in zT values in the range of 1.1–1.2. Recently, Lai et al. [57.50] combined density functional theory (DFT ) calculations with high resolution x-ray scattering studies to show that Sb lone pairs residing above and below the trigonal Cu–S plane (Fig. 57.9) act to displace the Cu away from the plane, causing a buckling-type behavior. In this picture, the three sulfur atoms surrounding the copper ion together with the lone-pair-containing antimony above and below the plane comprise a bipyramidal cage in which the copper atom rattles. This PGEC-like behavior is thought to be the source of low thermal conductivity in these compounds.
Fig. 57.9

The trigonally coordinated Cu site in tetrahedrite. A copper atom (gray) is trigonally surrounded by three sulfur atoms (yellow). Two antimony atoms (brown) possessing unbonded lone pair electrons (black dots) reside above and below the trigonal plane. The Sb lone pairs act to pull the copper atom of the trigonal plane. The trigonal bipyramid arrangement is a cage in which the Cu atom rattles, giving rise to PGEC-like behavior

With the growing ability to calculate the properties of materials from first principles, approaches such as that used for tetrahedrite offer the opportunity to use powerful materials-by-design principles to conceive of new structures and compositions with the unusual combination of transport properties required for thermoelectricity directly built-in to them. This is a new paradigm in thermoelectric materials research that offers great promise from thermoelectric in the future.

57.4 Summary and Future Outlook

Thermoelectrics are very unique material systems in which the fundamental behavior associated with electrons and phonons in crystalline solids can give rise to interesting thermal and electronic effects, which can be important technologically. The stringent requirements on Seebeck coefficient, electrical conductivity, and thermal conductivity that are necessary to produce good thermoelectric behavior continue to offer challenges and inspiration to myriad researchers. Thermoelectricity has become a proving ground for numerous new and innovative concepts in physics, chemistry, and materials science, including quantum size effects, nanostructuring, phonon-glass-electron crystal behavior, and large anharmonicity. New knowledge uncovered in these and other areas is driving improvements in material performance, and the future holds promise that highly efficient thermoelectric devices for both power generation from waste heat and solid-state climate control will become a reality.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Chemical Engineering and Materials ScienceMichigan State UniversityEast LansingUSA

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