Electronic band structure

From Wikipedia, the free encyclopedia

In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energies that an electron within the solid may have (called energy bands, allowed bands, or simply bands) and ranges of energy that it may not have (called band gaps or forbidden bands). 

 

Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as electrical resistivity and optical absorption, and forms the foundation of the understanding of all solid-state devices (transistors, solar cells, etc.).

Why bands and band gaps occur

Showing how electronic band structure comes about by the hypothetical example of a large number of carbon atoms being brought together to form a diamond crystal. The graph (right) shows the energy levels as a function of the spacing between atoms. When the atoms are far apart (right side of graph) each atom has valence atomic orbitals p and s which have the same energy. However when the atoms come closer together their orbitals begin to overlap. Due to the Pauli Exclusion Principle each atomic orbital splits into N molecular orbitals each with a different energy, where N is the number of atoms in the crystal. Since N is such a large number, adjacent orbitals are extremely close together in energy so the orbitals can be considered a continuous energy band. a is the atomic spacing in an actual crystal of diamond. At that spacing the orbitals form two bands, called the valence and conduction bands, with a 5.5 eV band gap between them.

 

The electrons of a single, isolated atom occupy atomic orbitals each of which has a discrete energy level. When two or more atoms join together to form into a molecule, their atomic orbitals overlap. The Pauli exclusion principle dictates that no two electrons can have the same quantum numbers in a molecule. So if two identical atoms combine to form a diatomic molecule, each atomic orbital splits into two molecular orbitals of different energy, allowing the electrons in the former atomic orbitals to occupy the new orbital structure without any having the same energy. 

Similarly if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap. Since the Pauli exclusion principle dictates that no two electrons in the solid have the same quantum numbers, each atomic orbital splits into N discrete molecular orbitals, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (N~10²²) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10⁻²² eV). The energy of adjacent levels is so close together that they can be considered as a continuum, an energy band.

This formation of bands is mostly a feature of the outermost electrons (valence electrons) in the atom, which are the ones involved in chemical bonding and electrical conductivity. The inner electron orbitals do not overlap to a significant degree, so their bands are very narrow.

Band gaps are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. The bands have different widths, with the widths depending upon the degree of overlap in the atomic orbitals from which they arise. Two adjacent bands may simply not be wide enough to fully cover the range of energy. For example, the bands associated with core orbitals (such as 1s electrons) are extremely narrow due to the small overlap between adjacent atoms. As a result, there tend to be large band gaps between the core bands. Higher bands involve comparatively larger orbitals with more overlap, becoming progressively wider at higher energies so that there are no band gaps at higher energies.

Basic concepts

Assumptions and limits of band structure theory

Band theory is only an approximation to the quantum state of a solid, which applies to solids consisting of many identical atoms or molecules bonded together. These are the assumptions necessary for band theory to be valid:

• Infinite-size system: For the bands to be continuous, the piece of material must consist of a large number of atoms. Since a macroscopic piece of material contains on the order of 10²² atoms, this is not a serious restriction; band theory even applies to microscopic-sized transistors in integrated circuits. With modifications, the concept of band structure can also be extended to systems which are only "large" along some dimensions, such as two-dimensional electron systems.
• Homogeneous system: Band structure is an intrinsic property of a material, which assumes that the material is homogeneous. Practically, this means that the chemical makeup of the material must be uniform throughout the piece.
• Non-interactivity: The band structure describes "single electron states". The existence of these states assumes that the electrons travel in a static potential without dynamically interacting with lattice vibrations, other electrons, photons, etc.

The above assumptions are broken in a number of important practical situations, and the use of band structure requires one to keep a close check on the limitations of band theory:

• Inhomogeneities and interfaces: Near surfaces, junctions, and other inhomogeneities, the bulk band structure is disrupted. Not only are there local small-scale disruptions (e.g., surface states or dopant states inside the band gap), but also local charge imbalances. These charge imbalances have electrostatic effects that extend deeply into semiconductors, insulators, and the vacuum.
• Along the same lines, most electronic effects (capacitance, electrical conductance, electric-field screening) involve the physics of electrons passing through surfaces and/or near interfaces. The full description of these effects, in a band structure picture, requires at least a rudimentary model of electron-electron interactions.
• Small systems: For systems which are small along every dimension (e.g., a small molecule or a quantum dot), there is no continuous band structure. The crossover between small and large dimensions is the realm of mesoscopic physics.
• Strongly correlated materials (for example, Mott insulators) simply cannot be understood in terms of single-electron states. The electronic band structures of these materials are poorly defined (or at least, not uniquely defined) and may not provide useful information about their physical state.

Crystalline symmetry and wavevectors

Fig 1. Brillouin zone of a face-centered cubic lattice showing labels for special symmetry points.

 

Fig 2. Band structure plot for Si, Ge, GaAs and InAs generated with tight binding model. Note that Si and Ge are indirect band gap materials, while GaAs and InAs are direct.

Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The single-electron Schrödinger equation is solved for an electron in a lattice-periodic potential, giving Bloch waves as solutions:

${\displaystyle \psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} )}$,

where k is called the wavevector. For each value of k, there are multiple solutions to the Schrödinger equation labelled by n, the band index, which simply numbers the energy bands. Each of these energy levels evolves smoothly with changes in k, forming a smooth band of states. For each band we can define a function E_n(k), which is the dispersion relation for electrons in that band. 

The wavevector takes on any value inside the Brillouin zone, which is a polyhedron in wavevector space that is related to the crystal's lattice. Wavevectors outside the Brillouin zone simply correspond to states that are physically identical to those states within the Brillouin zone. Special high symmetry points/lines in the Brillouin zone are assigned labels like Γ, Δ, Λ, Σ (see Fig 1).

It is difficult to visualize the shape of a band as a function of wavevector, as it would require a plot in four-dimensional space, E vs. k_x, k_y, k_z. In scientific literature it is common to see band structure plots which show the values of E_n(k) for values of k along straight lines connecting symmetry points, often labelled Δ, Λ, Σ respectively. Another method for visualizing band structure is to plot a constant-energy isosurface in wavevector space, showing all of the states with energy equal to a particular value. The isosurface of states with energy equal to the Fermi level is known as the Fermi surface. 

Energy band gaps can be classified using the wavevectors of the states surrounding the band gap:

• Direct band gap: the lowest-energy state above the band gap has the same k as the highest-energy state beneath the band gap.
• Indirect band gap: the closest states above and beneath the band gap do not have the same k value.

Asymmetry: Band structures in non-crystalline solids

Although electronic band structures are usually associated with crystalline materials, quasi-crystalline and amorphous solids may also exhibit band structures. These are somewhat more difficult to study theoretically since they lack the simple symmetry of a crystal, and it is not usually possible to determine a precise dispersion relation. As a result, virtually all of the existing theoretical work on the electronic band structure of solids has focused on crystalline materials.

Density of states

The density of states function g(E) is defined as the number of electronic states per unit volume, per unit energy, for electron energies near E. 

The density of states function is important for calculations of effects based on band theory. In Fermi's Golden Rule, a calculation for the rate of optical absorption, it provides both the number of excitable electrons and the number of final states for an electron. It appears in calculations of electrical conductivity where it provides the number of mobile states, and in computing electron scattering rates where it provides the number of final states after scattering.

For energies inside a band gap, g(E) = 0.

Filling of bands

Filling of the electronic states in various types of materials at equilibrium. Here, height is energy while width is the density of available states for a certain energy in the material listed. The shade follows the Fermi–Dirac distribution (black = all states filled, white = no state filled). In metals and semimetals the Fermi level E_F lies inside at least one band. In insulators and semiconductors the Fermi level is inside a band gap; however, in semiconductors the bands are near enough to the Fermi level to be thermally populated with electrons or holes. 

 

At thermodynamic equilibrium, the likelihood of a state of energy E being filled with an electron is given by the Fermi–Dirac distribution, a thermodynamic distribution that takes into account the Pauli exclusion principle:

${\displaystyle f(E)={\frac {1}{1+e^{{(E-\mu )}/{k_{\rm {B}}T}}}}}$

where:

• k_BT is the product of Boltzmann's constant and temperature, and
• µ is the total chemical potential of electrons, or Fermi level (in semiconductor physics, this quantity is more often denoted E_F). The Fermi level of a solid is directly related to the voltage on that solid, as measured with a voltmeter. Conventionally, in band structure plots the Fermi level is taken to be the zero of energy (an arbitrary choice).

The density of electrons in the material is simply the integral of the Fermi–Dirac distribution times the density of states:

${\displaystyle N/V=\int _{-\infty }^{\infty }g(E)f(E)\,dE}$

Although there are an infinite number of bands and thus an infinite number of states, there are only a finite number of electrons to place in these bands. The preferred value for the number of electrons is a consequence of electrostatics: even though the surface of a material can be charged, the internal bulk of a material prefers to be charge neutral. The condition of charge neutrality means that N/V must match the density of protons in the material. For this to occur, the material electrostatically adjusts itself, shifting its band structure up or down in energy (thereby shifting g(E)), until it is at the correct equilibrium with respect to the Fermi level.

Names of bands near the Fermi level (conduction band, valence band)

A solid has an infinite number of allowed bands, just as an atom has infinitely many energy levels. However, most of the bands simply have too high energy, and are usually disregarded under ordinary circumstances. Conversely, there are very low energy bands associated with the core orbitals (such as 1s electrons). These low-energy core bands are also usually disregarded since they remain filled with electrons at all times, and are therefore inert. Likewise, materials have several band gaps throughout their band structure. 

The most important bands and band gaps—those relevant for electronics and optoelectronics—are those with energies near the Fermi level. The bands and band gaps near the Fermi level are given special names, depending on the material:

• In a semiconductor or band insulator, the Fermi level is surrounded by a band gap, referred to as the band gap (to distinguish it from the other band gaps in the band structure). The closest band above the band gap is called the conduction band, and the closest band beneath the band gap is called the valence band. The name "valence band" was coined by analogy to chemistry, since in semiconductors (and insulators) the valence band is built out of the valence orbitals.
• In a metal or semimetal, the Fermi level is inside of one or more allowed bands. In semimetals the bands are usually referred to as "conduction band" or "valence band" depending on whether the charge transport is more electron-like or hole-like, by analogy to semiconductors. In many metals, however, the bands are neither electron-like nor hole-like, and often just called "valence band" as they are made of valence orbitals. The band gaps in a metal's band structure are not important for low energy physics, since they are too far from the Fermi level.

Theory in crystals

The ansatz is the special case of electron waves in a periodic crystal lattice using Bloch waves as treated generally in the dynamical theory of diffraction. Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b₁,b₂,b₃). Now, any periodic potential V(r) which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:

${\displaystyle V(\mathbf {r} )=\sum _{\mathbf {K} }{V_{\mathbf {K} }e^{i\mathbf {K} \cdot \mathbf {r} }}}$

where K = m₁b₁ + m₂b₂ + m₃b₃ for any set of integers (m₁,m₂,m₃). 

From this theory, an attempt can be made to predict the band structure of a particular material, however most ab initio methods for electronic structure calculations fail to predict the observed band gap.

Nearly free electron approximation

In the nearly free electron approximation, interactions between electrons are completely ignored. This approximation allows use of Bloch's Theorem which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors. The consequences of periodicity are described mathematically by the Bloch wavefunction:

${\displaystyle {\Psi }_{n,\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n}(\mathbf {r} )}$

where the function ${\displaystyle u_{n}(\mathbf {r} )}$ is periodic over the crystal lattice, that is,

${\displaystyle u_{n}(\mathbf {r} )=u_{n}(\mathbf {r-R} )}$.

Here index n refers to the n-th energy band, wavevector k is related to the direction of motion of the electron, r is the position in the crystal, and R is the location of an atomic site.

The NFE model works particularly well in materials like metals where distances between neighbouring atoms are small. In such materials the overlap of atomic orbitals and potentials on neighbouring atoms is relatively large. In that case the wave function of the electron can be approximated by a (modified) plane wave. The band structure of a metal like aluminium even gets close to the empty lattice approximation.

Tight binding model

The opposite extreme to the nearly free electron approximation assumes the electrons in the crystal behave much like an assembly of constituent atoms. This tight binding model assumes the solution to the time-independent single electron Schrödinger equation ${\displaystyle \Psi }$ is well approximated by a linear combination of atomic orbitals ${\displaystyle \psi _{n}(\mathbf {r} )}$.

${\displaystyle \Psi (\mathbf {r} )=\sum _{n,\mathbf {R} }b_{n,\mathbf {R} }\psi _{n}(\mathbf {r-R} )}$,

where the coefficients ${\displaystyle b_{n,\mathbf {R} }}$ are selected to give the best approximate solution of this form. Index n refers to an atomic energy level and R refers to an atomic site. A more accurate approach using this idea employs Wannier functions, defined by:

${\displaystyle a_{n}(\mathbf {r-R} )={\frac {V_{C}}{(2\pi )^{3}}}\int \limits _{\text{BZ}}d\mathbf {k} e^{-i\mathbf {k} \cdot (\mathbf {R-r} )}u_{n\mathbf {k} }}$;

in which ${\displaystyle u_{n\mathbf {k} }}$ is the periodic part of the Bloch wave and the integral is over the Brillouin zone. Here index n refers to the n-th energy band in the crystal. The Wannier functions are localized near atomic sites, like atomic orbitals, but being defined in terms of Bloch functions they are accurately related to solutions based upon the crystal potential. Wannier functions on different atomic sites R are orthogonal. The Wannier functions can be used to form the Schrödinger solution for the n-th energy band as:

${\displaystyle \Psi _{n,\mathbf {k} }(\mathbf {r} )=\sum _{\mathbf {R} }e^{-i\mathbf {k} \cdot (\mathbf {R-r} )}a_{n}(\mathbf {r-R} )}$.

The TB model works well in materials with limited overlap between atomic orbitals and potentials on neighbouring atoms. Band structures of materials like Si, GaAs, SiO₂ and diamond for instance are well described by TB-Hamiltonians on the basis of atomic sp³ orbitals. In transition metals a mixed TB-NFE model is used to describe the broad NFE conduction band and the narrow embedded TB d-bands. The radial functions of the atomic orbital part of the Wannier functions are most easily calculated by the use of pseudopotential methods. NFE, TB or combined NFE-TB band structure calculations, sometimes extended with wave function approximations based on pseudopotential methods, are often used as an economic starting point for further calculations.

KKR model

The simplest form of this approximation centers non-overlapping spheres (referred to as muffin tins) on the atomic positions. Within these regions, the potential experienced by an electron is approximated to be spherically symmetric about the given nucleus. In the remaining interstitial region, the screened potential is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.

A variational implementation was suggested by Korringa and by Kohn and Rostocker, and is often referred to as the KKR model.

Density-functional theory

In recent physics literature, a large majority of the electronic structures and band plots are calculated using density-functional theory (DFT), which is not a model but rather a theory, i.e., a microscopic first-principles theory of condensed matter physics that tries to cope with the electron-electron many-body problem via the introduction of an exchange-correlation term in the functional of the electronic density. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example by angle-resolved photoemission spectroscopy (ARPES). In particular, the band shape is typically well reproduced by DFT. But there are also systematic errors in DFT bands when compared to experiment results. In particular, DFT seems to systematically underestimate by about 30-40% the band gap in insulators and semiconductors.

It is commonly believed that DFT is a theory to predict ground state properties of a system only (e.g. the total energy, the atomic structure, etc.), and that excited state properties cannot be determined by DFT. This is a misconception. In principle, DFT can determine any property (ground state or excited state) of a system given a functional that maps the ground state density to that property. This is the essence of the Hohenberg–Kohn theorem. In practice, however, no known functional exists that maps the ground state density to excitation energies of electrons within a material. Thus, what in the literature is quoted as a DFT band plot is a representation of the DFT Kohn–Sham energies, i.e., the energies of a fictive non-interacting system, the Kohn–Sham system, which has no physical interpretation at all. The Kohn–Sham electronic structure must not be confused with the real, quasiparticle electronic structure of a system, and there is no Koopman's theorem holding for Kohn–Sham energies, as there is for Hartree–Fock energies, which can be truly considered as an approximation for quasiparticle energies. Hence, in principle, Kohn–Sham based DFT is not a band theory, i.e., not a theory suitable for calculating bands and band-plots. In principle time-dependent DFT can be used to calculate the true band structure although in practice this is often difficult. A popular approach is the use of hybrid functionals, which incorporate a portion of Hartree–Fock exact exchange; this produces a substantial improvement in predicted bandgaps of semiconductors, but is less reliable for metals and wide-bandgap materials.

Green's function methods and the ab initio GW approximation

To calculate the bands including electron-electron interaction many-body effects, one can resort to so-called Green's function methods. Indeed, knowledge of the Green's function of a system provides both ground (the total energy) and also excited state observables of the system. The poles of the Green's function are the quasiparticle energies, the bands of a solid. The Green's function can be calculated by solving the Dyson equation once the self-energy of the system is known. For real systems like solids, the self-energy is a very complex quantity and usually approximations are needed to solve the problem. One such approximation is the GW approximation, so called from the mathematical form the self-energy takes as the product Σ = GW of the Green's function G and the dynamically screened interaction W. This approach is more pertinent when addressing the calculation of band plots (and also quantities beyond, such as the spectral function) and can also be formulated in a completely ab initio way. The GW approximation seems to provide band gaps of insulators and semiconductors in agreement with experiment, and hence to correct the systematic DFT underestimation.

Mott insulators

Although the nearly free electron approximation is able to describe many properties of electron band structures, one consequence of this theory is that it predicts the same number of electrons in each unit cell. If the number of electrons is odd, we would then expect that there is an unpaired electron in each unit cell, and thus that the valence band is not fully occupied, making the material a conductor. However, materials such as CoO that have an odd number of electrons per unit cell are insulators, in direct conflict with this result. This kind of material is known as a Mott insulator, and requires inclusion of detailed electron-electron interactions (treated only as an averaged effect on the crystal potential in band theory) to explain the discrepancy. The Hubbard model is an approximate theory that can include these interactions. It can be treated non-perturbatively within the so-called dynamical mean field theory, which attempts to bridge the gap between the nearly free electron approximation and the atomic limit. Formally, however, the states are not non-interacting in this case and the concept of a band structure is not adequate to describe these cases.

Others

Calculating band structures is an important topic in theoretical solid state physics. In addition to the models mentioned above, other models include the following:

• Empty lattice approximation: the "band structure" of a region of free space that has been divided into a lattice.
• k·p perturbation theory is a technique that allows a band structure to be approximately described in terms of just a few parameters. The technique is commonly used for semiconductors, and the parameters in the model are often determined by experiment.
• The Kronig-Penney Model, a one-dimensional rectangular well model useful for illustration of band formation. While simple, it predicts many important phenomena, but is not quantitative.
• Hubbard model

The band structure has been generalised to wavevectors that are complex numbers, resulting in what is called a complex band structure, which is of interest at surfaces and interfaces. 

Each model describes some types of solids very well, and others poorly. The nearly free electron model works well for metals, but poorly for non-metals. The tight binding model is extremely accurate for ionic insulators, such as metal halide salts (e.g. NaCl).

Band diagrams

To understand how band structure changes relative to the Fermi level in real space, a band structure plot is often first simplified in the form of a band diagram. In a band diagram the vertical axis is energy while the horizontal axis represents real space. Horizontal lines represent energy levels, while blocks represent energy bands. When the horizontal lines in these diagram are slanted then the energy of the level or band changes with distance. Diagrammatically, this depicts the presence of an electric field within the crystal system. Band diagrams are useful in relating the general band structure properties of different materials to one another when placed in contact with each other.

Thermoelectric materials

From Wikipedia, the free encyclopedia

Thermoelectric materials show the thermoelectric effect in a strong or convenient form.

 

The thermoelectric effect refers to phenomena by which either a temperature difference creates an electric potential or an electric potential creates a temperature difference. These phenomena are known more specifically as the Seebeck effect (converting temperature to current), Peltier effect (converting current to temperature), and Thomson effect (conductor heating/cooling). While all materials have a nonzero thermoelectric effect, in most materials it is too small to be useful. However, low-cost materials that have a sufficiently strong thermoelectric effect (and other required properties) could be used in applications including power generation and refrigeration. A commonly used thermoelectric material in such applications is bismuth telluride (Bi
₂Te
₃).

Thermoelectric materials are used in thermoelectric systems for cooling or heating in niche applications, and are being studied as a way to regenerate electricity from waste heat.

Materials selection criteria

The usefulness of a material in thermoelectric systems is determined by the two factors device efficiency and power factor. These are determined by the material's electrical conductivity, thermal conductivity, Seebeck coefficient and behavior under changing temperatures.

Device efficiency

The efficiency of a thermoelectric device for electricity generation is given by ${\displaystyle \eta }$, defined as

${\displaystyle \eta ={{\text{energy provided to the load}} \over {\text{heat energy absorbed at hot junction}}}.}$

The ability of a given material to efficiently produce thermoelectric power is related to its dimensionless figure of merit given by

${\displaystyle ZT={\sigma S^{2}T \over \kappa }}$,

which depends on the Seebeck coefficient S, thermal conductivity κ, electrical conductivity σ, and temperature T. 

In an actual thermoelectric device, two materials are used. The maximum efficiency ${\displaystyle \eta _{\mathrm {max} }}$ is then given by

${\displaystyle \eta _{\mathrm {max} }={T_{H}-T_{C} \over T_{H}}{{\sqrt {1+Z{\bar {T}}}}-1 \over {\sqrt {1+Z{\bar {T}}}}+{T_{C} \over T_{H}}},}$

where ${\displaystyle T_{H}}$ is the temperature at the hot junction and ${\displaystyle T_{C}}$ is the temperature at the surface being cooled. ${\displaystyle Z{\bar {T}}}$ is the modified dimensionless figure of merit, which takes into consideration the thermoelectric capacity of both thermoelectric materials being used in the device and, after geometrical optimization regarding the legs sections, is defined as

${\displaystyle Z{\bar {T}}={(S_{p}-S_{n})^{2}{\bar {T}} \over [(\rho _{n}\kappa _{n})^{1/2}+(\rho _{p}\kappa _{p})^{1/2}]^{2}}}$

where ${\displaystyle \rho }$ is the electrical resistivity, ${\displaystyle {\bar {T}}}$ is the average temperature between the hot and cold surfaces and the subscripts n and p denote properties related to the n- and p-type semiconducting thermoelectric materials, respectively. Since thermoelectric devices are heat engines, their efficiency is limited by the Carnot efficiency, hence the ${\displaystyle T_{H}}$ and ${\displaystyle T_{C}}$ terms in ${\displaystyle \eta _{\mathrm {max} }}$. Regardless, the coefficient of performance of current commercial thermoelectric refrigerators ranges from 0.3 to 0.6, one-sixth the value of traditional vapor-compression refrigerators.

Power factor

To determine the usefulness of a material in a thermoelectric generator or a thermoelectric cooler the power factor is calculated by its Seebeck coefficient and its electrical conductivity under a given temperature difference:

${\displaystyle \mathrm {Power~factor} =\sigma S^{2}.}$

where S is the Seebeck coefficient, and σ is the electrical conductivity. 

Materials with a high power factor are able to 'generate' more energy (move more heat or extract more energy from that temperature difference) in a space-constrained application, but are not necessarily more efficient in generating this energy.

Aspects of materials choice

For good efficiency, materials with high electrical conductivity, low thermal conductivity and high Seebeck coefficient are needed.

State density: metals vs semiconductors

The band structure of semiconductors offers better thermoelectric effects than the band structure of metals. 

The Fermi energy is below the conduction band causing the state density to be asymmetric around the Fermi energy. Therefore, the average electron energy of the conduction band is higher than the Fermi energy, making the system conducive for charge motion into a lower energy state. By contrast, the Fermi energy lies in the conduction band in metals. This makes the state density symmetric about the Fermi energy so that the average conduction electron energy is close to the Fermi energy, reducing the forces pushing for charge transport. Therefore, semiconductors are ideal thermoelectric materials. Due to the small Seebeck coefficient metals have a very limited performance and the main materials of interest are Semiconductors.

Conductivity

In the efficiency equations above, thermal conductivity and electrical conductivity compete.

The thermal conductivity κ has mainly two components:

κ = κ _electron + κ _phonon

According to the Wiedemann–Franz law, the higher the electrical conductivity, the higher κ _electron becomes. Thus in metals the ratio of thermal to electrical conductivity is about fixed, as the electron part dominates. In semiconductors, the phonon part is important and can not be neglected. It reduces the efficiency. For good efficiency a low ratio of κ _phonon / κ _electron is desired. 

Therefore, it is necessary to minimize κ _phonon and keep the electrical conductivity high. Thus semiconductors should be highly doped. 

G. A. Slack proposed that in order to optimize the figure of merit, phonons, which are responsible for thermal conductivity must experience the material as a glass (experiencing a high degree of phonon scattering—lowering thermal conductivity) while electrons must experience it as a crystal (experiencing very little scattering—maintaining electrical conductivity). The figure of merit can be improved through the independent adjustment of these properties.

Quality Factor (detailed theory on semiconductors)

The maximum ${\displaystyle Z{\bar {T}}}$ of a material is given by the material's Quality Factor: 

${\displaystyle B={\frac {2k_{B}^{2}\hbar }{3\pi }}{\frac {N_{v}C_{l}}{m_{l}^{*}\Xi ^{2}\kappa _{L}}}T}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle N_{v}}$ is the number of degenerated valleys for the band, ${\displaystyle C_{l}}$ is the average longitudinal elastic moduli, ${\displaystyle m_{l}^{*}}$ is the inertial effective mass, ${\displaystyle \Xi }$ is the deformation potential coefficient, ${\displaystyle \kappa _{L}}$ is the lattice thermal conduction, and ${\displaystyle T}$ is temperature. The figure of merit, ${\displaystyle Z{\bar {T}}}$, depends on doping concentration and temperature of the material of interest. The material Quality Factor: ${\displaystyle B}$ is useful because it allows for an intrinsic comparisons of possible efficiency between different materials. This relation shows that improving the electronic component ${\displaystyle {\frac {N_{v}}{m_{l}^{*}\Xi ^{2}}}}$, which primarily affects the Seebeck coefficient, will increase the quality factor of a material. A large density of states can be created due to a large number of conducting bands (${\displaystyle N_{v}}$) or by flat bands giving a high band effective mass (${\displaystyle m_{b}^{*}}$). For isotropic materials ${\displaystyle m_{b}^{*}=m_{l}^{*}}$. Therefore, it is desirable for thermoelectric materials to have high valley degeneracy in a very sharp band structure. Other complex features of the electronic structure are important. These can be partially quantified using an electronic fitness function.

Materials of interest

Strategies to improve thermoelectrics include both advanced bulk materials and the use of low-dimensional systems. Such approaches to reduce lattice thermal conductivity fall under three general material types: (1) Alloys: create point defects, vacancies, or rattling structures (heavy-ion species with large vibrational amplitudes contained within partially filled structural sites) to scatter phonons within the unit cell crystal; (2) Complex crystals: separate the phonon glass from the electron crystal using approaches similar to those for superconductors (the region responsible for electron transport should be an electron crystal of a high-mobility semiconductor, while the phonon glass should ideally house disordered structures and dopants without disrupting the electron crystal, analogous to the charge reservoir in high-T_c superconductors); (3) Multiphase nanocomposites: scatter phonons at the interfaces of nanostructured materials, be they mixed composites or thin film superlattices. 

Materials under consideration for thermoelectric device applications include:

Bismuth chalcogenides and their nanostructures

Materials such as Bi
₂Te
₃ and Bi
₂Se
₃ comprise some of the best performing room temperature thermoelectrics with a temperature-independent figure-of-merit, ZT, between 0.8 and 1.0. Nanostructuring these materials to produce a layered superlattice structure of alternating Bi
₂Te
₃ and Sb
₂Te
₃ layers produces a device within which there is good electrical conductivity but perpendicular to which thermal conductivity is poor. The result is an enhanced ZT (approximately 2.4 at room temperature for p-type). Note that this high value of ZT has not been independently confirmed due to the complicated demands on the growth of such superlattices and device fabrication; however the material ZT values are consistent with the performance of hot-spot coolers made out of these materials and validated at Intel Labs. 

Bismuth telluride and its solid solutions are good thermoelectric materials at room temperature and therefore suitable for refrigeration applications around 300 K. The Czochralski method has been used to grow single crystalline bismuth telluride compounds. These compounds are usually obtained with directional solidification from melt or powder metallurgy processes. Materials produced with these methods have lower efficiency than single crystalline ones due to the random orientation of crystal grains, but their mechanical properties are superior and the sensitivity to structural defects and impurities is lower due to high optimal carrier concentration.

The required carrier concentration is obtained by choosing a nonstoichiometric composition, which is achieved by introducing excess bismuth or tellurium atoms to primary melt or by dopant impurities. Some possible dopants are halogens and group IV and V atoms. Due to the small bandgap (0.16 eV) Bi₂Te₃ is partially degenerate and the corresponding Fermi-level should be close to the conduction band minimum at room temperature. The size of the band-gap means that Bi₂Te₃ has high intrinsic carrier concentration. Therefore, minority carrier conduction cannot be neglected for small stoichiometric deviations. Use of telluride compounds is limited by the toxicity and rarity of tellurium.

Lead telluride

Heremans et al. (2008) demonstrated that thallium-doped lead telluride alloy (PbTe) achieves a ZT of 1.5 at 773 K. Later, Snyder et al. (2011) reported ZT~1.4 at 750 K in sodium-doped PbTe, and ZT~1.8 at 850 K in sodium-doped PbTe_1−xSe_x alloy. Snyder's group determined that both thallium and sodium alter the electronic structure of the crystal increasing electronic conductivity. They also claim that selenium increases electric conductivity and reduces thermal conductivity. 

In 2012 another team used lead telluride to convert 15 to 20 percent of waste heat to electricity, reaching a ZT of 2.2, which they claimed was the highest yet reported.

Inorganic clathrates

Inorganic clathrates have the general formula A_xB_yC_46-y (type I) and A_xB_yC_136-y (type II), where B and C are group III and IV elements, respectively, which form the framework where “guest” A atoms (alkali or alkaline earth metal) are encapsulated in two different polyhedra facing each other. The differences between types I and II come from the number and size of voids present in their unit cells. Transport properties depend on the framework's properties, but tuning is possible by changing the “guest” atoms.

The most direct approach to synthesize and optimize the thermoelectric properties of semiconducting type I clathrates is substitutional doping, where some framework atoms are replaced with dopant atoms. In addition, powder metallurgical and crystal growth techniques have been used in clathrate synthesis. The structural and chemical properties of clathrates enable the optimization of their transport properties as a function of stoichiometry. The structure of type II materials allows a partial filling of the polyhedra, enabling better tuning of the electrical properties and therefore better control of the doping level. Partially filled variants can be synthesized as semiconducting or even insulating.

Blake et al. have predicted ZT~0.5 at room temperature and ZT~1.7 at 800 K for optimized compositions. Kuznetsov et al. measured electrical resistance and Seebeck coefficient for three different type I clathrates above room temperature and by estimating high temperature thermal conductivity from the published low temperature data they obtained ZT~0.7 at 700 K for Ba₈Ga₁₆Ge₃₀ and ZT~0.87 at 870 K for Ba₈Ga₁₆Si₃₀.

Magnesium group IV compounds

Mg₂B^IV (B^IV=Si, Ge, Sn) compounds and their solid solutions are good thermoelectric materials and their ZT values are comparable with those of established materials. The appropriate production methods are based on direct co-melting, but mechanical alloying has also been used. During synthesis, magnesium losses due to evaporation and segregation of components (especially for Mg₂Sn) need to be taken into account. Directed crystallization methods can produce single crystalline material. Solid solutions and doped compounds have to be annealed in order to produce homogeneous samples - with the same properties throughout. At 800 K, Mg₂Si_0.55−xSn_0.4Ge_0.05Bi_x has been reported to have a figure of merit about 1.4, the highest ever reported for these compounds.

Silicides

Higher silicides display ZT levels with current materials. They are mechanically and chemically strong and therefore can often be used in harsh environments without protection. Possible fabrication methods include Czochralski and floating zone for single crystals and hot pressing and sintering for polycrystalline.

Skutterudite thermoelectrics

Recently, skutterudite materials have sparked the interest of researchers in search of new thermoelectrics. These structures are of the form (Co,Ni,Fe)(P,Sb,As)
₃ and are cubic with space group Im3. Unfilled, these materials contain voids into which low-coordination ions (usually rare-earth elements) can be inserted in order to alter thermal conductivity by producing sources for lattice phonon scattering and decrease thermal conductivity due to the lattice without reducing electrical conductivity. Such qualities make these materials exhibit PGEC behavior. However, recently Khan et al. (2017) showed that it is possible to reduce the thermal conductivity without filling these voids and enhance the figure of merit by 100%, with special architecture containing nano and micro pores.

Skutterudites have the chemical formula LM₄X₁₂, where L is a rare-earth metal, M a transition metal and X a metalloid, a group V element or pnictogen whose properties lie between those of a metal and nonmetal such as phosphorus, antimony, or arsenic. These materials could be potential in multistage thermoelectric devices as it has been shown that they have ZT>1.0, but their properties are not well known.

Oxide thermoelectrics

Homologous oxide compounds (such as those of the form (SrTiO
₃)_n(SrO)
_m—the Ruddlesden-Popper phase) have layered superlattice structures that make them promising candidates for use in high-temperature thermoelectric devices. These materials exhibit low thermal conductivity perpendicular to the layers while maintaining good electronic conductivity within the layers. ZT values are relatively low (~0.34 at 1,000K), but their enhanced thermal stability, as compared to conventional high-ZT bismuth compounds, makes them superior for use in high-temperature applications.

Interest in oxides as thermoelectric materials was reawakened in 1997 when Na_xCoO₂ was found to exhibit good thermoelectric behavior. In addition to their thermal stability, other advantages of oxides are their non-toxicity and high oxidation resistance. Simultaneously controlling both the electric and phonon systems may require nanostructured materials. Some layered oxide materials are thought to have ZT~2.7 at 900 K. If the layers in a given material have the same stoichiometry, they will be stacked so that the same atoms will not be positioned on top of each other, impeding phonon conductivity perpendicular to the layers. Recently, oxide thermoelectrics have gained a lot of attention so that the range of promising phases increased drastically. Novel members of this family include ZnO, MnO₂, and NbO₂, to name but a few.

Half Heusler alloys

Half Heusler alloys have a great potential for high- temperature power generation applications. Half-Heusler (HH) are alloys with a Formula ABX . Examples of Half-Heusler include NbFeSb, NbCoSn and VFeSb. HH possesses cubic MgAgAs type structure, forming three interpenetrating face-centered-cubic (FCC). The ability to substitute any of these three sublattices opens the door for wide variety of HH compounds to be synthesized. A and B sites substitutions are employed to reduce the thermal conductivity, while the X site substitution is used to enhance the carrier concentration and thus the electrical conductivity.

Previously, ZT could not peak more than 0.5 for p-type and 0.8 for n-type HH compound. However, in the past few years, researchers were able to achieve ZT≈1 for both n-type and p-type. Nano-sized grains is one of the approaches used to lower the thermal conductivity via grain boundaries- assisted phonon scattering. Other approach was to utilize the principles of nanocomposites, by which certain combination of metals were favored on others due to the atomic size difference. For instance, Hf and Ti is more effective than Hf and Zr, when reduction of thermal conductivity is of concern, since the atomic size difference between the former is larger than that of the latter.

Electrically conducting organic materials

Some electrically conducting organic materials may have a higher figure of merit than existing inorganic materials. Seebeck coefficient can be even millivolts per Kelvin but electrical conductivity is usually low, resulting in small ZT values. Quasi-one-dimensional organic crystals are formed from linear chains or stacks of molecules that are packed into a 3D crystal. Under certain conditions some Q1D organic crystals may have ZT~20 at room temperature for both p- and n-type materials. This has been credited to an unspecified interference between two main electron-phonon interactions leading to the formation of narrow strip of states in the conduction band with a significantly reduced scattering rate as the mechanism compensate each other, yielding high ZT.

Silicon-germanium

Silicon-germanium alloys are currently the best thermoelectric materials around 1000 ℃ and are therefore used in some radioisotope thermoelectric generators (RTG) (notably the MHW-RTG and GPHS-RTG) and some other high temperature applications, such as waste heat recovery. Usability of silicon-germanium alloys is limited by their price and mid-range ZT (~0.7).

Sodium cobaltate

Experiments on crystals of sodium cobaltate, using X-ray and neutron scattering experiments carried out at the European Synchrotron Radiation Facility (ESRF) and the Institut Laue-Langevin (ILL) in Grenoble were able to suppress thermal conductivity by a factor of six compared to vacancy-free sodium cobaltate. The experiments agreed with corresponding density functional calculations. The technique involved large anharmonic displacements of Na
_0.8CoO
₂ contained within the crystals.

Amorphous materials

In 2002, Nolas and Goldsmid have come up with a suggestion that systems with the phonon mean free path larger than the charge carrier mean free path can exhibit an enhanced thermoelectric efficiency. This can be realized in amorphous thermoelectrics and soon they became a focus of many studies. This ground-breaking idea was accomplished in Cu-Ge-Te, NbO₂, In-Ga-Zn-O, Zr-Ni-Sn, Si-Au, and Ti-Pb-V-O amorphous systems. It should be mentioned that modelling of transport properties is challenging enough without breaking the long-range order so that design of amorphous thermoelectrics is at its infancy. Naturally, amorphous thermoelectrics give rise to extensive phonon scattering, which is still a challenge for crystalline thermoelectrics. A bright future is expected for these materials.

Functionally graded materials

Functionally graded materials make it possible to improve the conversion efficiency of existing thermoelectrics. These materials have a non-uniform carrier concentration distribution and in some cases also solid solution composition. In power generation applications the temperature difference can be several hundred degrees and therefore devices made from homogeneous materials have some part that operates at the temperature where ZT is substantially lower than its maximum value. This problem can be solved by using materials whose transport properties vary along their length thus enabling substantial improvements to the operating efficiency over large temperature differences. This is possible with functionally graded materials as they have a variable carrier concentration along the length of the material, which is optimized for operations over specific temperature range.

Nanomaterials and superlattices

In addition to nanostructured Bi
₂Te
₃/Sb
₂Te
₃ superlattice thin films, other nanostructured materials, including nanowires, nanotubes and quantum dots show potential in improving thermoelectric properties.

PbTe/PbSeTe quantum dot superlattice

Another example of a superlattice involves a PbTe/PbSeTe quantum dot superlattices provides an enhanced ZT (approximately 1.5 at room temperature) that was higher than the bulk ZT value for either PbTe or PbSeTe (approximately 0.5).

Nanocrystal stability and thermal conductivity

Not all nanocrystalline materials are stable, because the crystal size can grow at high temperatures, ruining the materials' desired characteristics. 

Nanocrystalline materials have many interfaces between crystals, which Physics of SASER scatter phonons so the thermal conductivity is reduced. Phonons are confined to the grain, if their mean free path is larger than the material grain size. 

Measured lattice thermal conductivity in nanowires is known to depend on roughness, the method of synthesis and properties of the source material.

Nanocrystalline transition metal silicides

Nanocrystalline transition metal silicides are a promising material group for thermoelectric applications, because they fulfill several criteria that are demanded from the commercial applications point of view. In some nanocrystalline transition metal silicides the power factor is higher than in the corresponding polycrystalline material but the lack of reliable data on thermal conductivity prevents the evaluation of their thermoelectric efficiency.

Nanostructured skutterudites

Skutterudites, a cobalt arsenide mineral with variable amounts of nickel and iron, can be produced artificially, and are candidates for better thermoelectric materials. 

One advantage of nanostructured skutterudites over normal skutterudites is their reduced thermal conductivity, caused by grain boundary scattering. ZT values of ~0.65 and > 0.4 have been achieved with CoSb₃ based samples; the former values were 2.0 for Ni and 0.75 for Te-doped material at 680 K and latter for Au-composite at T > 700 K.

Even greater performance improvements can be achieved by using composites and by controlling the grain size, the compaction conditions of polycrystalline samples and the carrier concentration.

Graphene

Graphene is known for its high electrical conductivity and Seebeck coefficient at room temperature. However, from thermoelectric perspective, its thermal conductivity is notably high, which in turn limits its ZT. Several approaches were suggested to reduce the thermal conductivity of graphene without altering much its electrical conductivity. These include, but not limited to, the following:

1- Doping with carbon isotopes to form isotopic heterojunction such as that of 12C and 13C. Basically, those isotopes possess different phonon frequency mismatch, which ultimately lead to the scattering of the heat carriers (the phonons). Fortunately, this approach has been shown to not affecting neither the power factor nor the electrical conductivity.

2- Wrinkles and cracks in the graphene structure were shown to contribute to the reduction in the thermal conductivity. Reported values of thermal conductivity of suspended graphene of size 3.8 µm show a wide spread from 1500 to 5000 W/mK. A recent study attributed that to the microstructural defects present in graphene, such as wrinkles and cracks, which can drop the thermal conductivity by 27%. These defects help scatter phonons. 

3- Introduction of defects with techniques such as oxygen plasma treatment: a more systemic way of introducing defects in graphene structure is done through O₂ plasma treatment. Ultimately, the graphene sample will contain prescribed-holes spaced and numbered according to the plasma intensity. People were able to improve ZT of graphene from 1 to a value of 2.6 when the defect density is raised from 0.04 to 2.5 (this number is an index of defect density and usually understood when compared to the corresponding value of the un-treated graphene, 0.04 in our case). Nevertheless, this technique would lower the electrical conductivity as well, which can be kept unchanged if the plasma processing parameters are optimized.

4- Functionalization of graphene by oxygen: the thermal behavior of graphene oxide has not been investigated extensively as compared to its counterpart; graphene. However, it was shown theoretically by Density Functional Theory (DFT) model that adding oxygen into the lattice of graphene reduces a lot its thermal conductivity due to phonon scattering effect. Scattering of phonons result from both acoustic mismatch and reduced symmetry in graphene structure after doping with oxygen. The reduction of thermal conductivity can easily exceed 50% with this approach.

Superlattices and roughness

Superlattices - nano structured thermocouples, are considered a good candidate for better thermoelectric device manufacturing, with materials that can be used in manufacturing this structure.

Their production is expensive for general-use due to fabrication processes based on expensive thin-film growth methods. However, since the amount of thin-film materials required for device fabrication with superlattices, is so much less than thin-film materials in bulk thermoelectric materials (almost by a factor of 1/10,000) the long-term cost advantage is indeed favorable. 

This is particularly true given the limited availability of tellurium causing competing solar applications for thermoelectric coupling systems to rise.

Superlattice structures also allow the independent manipulation of transport parameters by adjusting the structure itself, enabling research for better understanding of the thermoelectric phenomena in nanoscale, and studying the phonon-blocking electron-transmitting structures - explaining the changes in electric field and conductivity due to the material's nano-structure.

Many strategies exist to decrease the superlattice thermal conductivity that are based on engineering of phonon transport. The thermal conductivity along the film plane and wire axis can be reduced by creating diffuse interface scattering and by reducing the interface separation distance, both which are caused by interface roughness.

Interface roughness can naturally occur or may be artificially induced. 

In nature, roughness is caused by the mixing of atoms of foreign elements. Artificial roughness can be created using various structure types, such as quantum dot interfaces and thin-films on step-covered substrates.

Problems in superlattices

Reduced electrical conductivity: reduced phonon-scattering interface structures often also exhibit a decrease in electrical conductivity. 

The thermal conductivity in the cross-plane direction of the lattice is usually very low, but depending on the type of superlattice, the thermoelectric coefficient may increase because of changes to the band structure.

Low thermal conductivity in superlattices is usually due to strong interface scattering of phonons. Minibands are caused by the lack of quantum confinement within a well. The mini-band structure depends on the superlattice period so that with a very short period (~1 nm) the band structure approaches the alloy limit and with a long period (≥ ~60 nm) minibands become so close to each other that they can be approximated with a continuum.

Superlatice structure countermeasures: counter measures can be taken which practically eliminate the problem of decreased electrical conductivity in a reduced phonon-scattering interface. These measures include the proper choice of superlattice structure, taking advantage of mini-band conduction across superlattices, and avoiding quantum-confinement. It has been shown that because electrons and phonons have different wavelengths, it is possible to engineer the structure in such a way that phonons are scattered more diffusely at the interface than electrons.

Phonon confinement countermeasures: another approach to overcome the decrease in electrical conductivity in reduced phonon-scattering structures is to increase phonon reflectivity and therefore decrease the thermal conductivity perpendicular to the interfaces. This can be achieved by increasing the mismatch between the materials in adjacent layers, including density, group velocity, specific heat, and the phonon-spectrum. 

Interface roughness causes diffuse phonon scattering, which either increases or decreases the phonon reflectivity at the interfaces. A mismatch between bulk dispersion relations confines phonons, and the confinement becomes more favorable as the difference in dispersion increases. 

The amount of confinement is currently unknown as only some models and experimental data exist. As with a previous method, the effects on the electrical conductivity have to be considered.

Attempts to Localize long wavelength phonons by aperiodic superlattices or composite superlattices with different periodicities have been made. In addition, defects, especially dislocations, can be used to reduce thermal conductivity in low dimensional systems.

Parasitic heat: parasitic heat conduction in the barrier layers could cause significant performance loss. It has been proposed but not tested that this can be overcome by choosing a certain correct distance between the quantum wells.

The Seebeck coefficient can change its sign in superlattice nanowires due to the existence of minigaps as Fermi energy varies. This indicates that superlattices can be tailored to exhibit n or p-type behavior by using the same dopants as those that are used for corresponding bulk materials by carefully controlling Fermi energy or the dopant concentration. With nanowire arrays, it is possible to exploit semimetal-semiconductor transition due to the quantum confinement and use materials that normally would not be good thermoelectric materials in bulk form. Such elements are for example bismuth. The Seebeck effect could also be used to determine the carrier concentration and Fermi energy in nanowires.

In quantum dot thermoelectrics, unconventional or nonband transport behavior (e.g. tunneling or hopping) is necessary to utilize their special electronic band structure in the transport direction. It is possible to achieve ZT>2 at elevated temperatures with quantum dot superlattices, but they are almost always unsuitable for mass production. 

However, in superlattices, where quantum-effects are not involved, with film thickness of only a few micrometers (µm) to about 15 µm, Bi₂Te₃/Sb₂Te₃ superlattice material has been made into high-performance microcoolers and other devices. The performance of hot-spot coolers are consistent with the reported ZT~2.4 of superlattice materials at 300 K.

Nanocomposites are promising material class for bulk thermoelectric devices, but several challenges have to be overcome to make them suitable for practical applications. It is not well understood why the improved thermoelectric properties appear only in certain materials with specific fabrication processes.

SrTe nanocrystals can be embedded in a bulk PbTe matrix so that rocksalt lattices of both materials are completely aligned (endotaxy) with optimal molar concentration for SrTe only 2%. This can cause strong phonon scattering but would not affect charge transport. In such case, ZT~1.7 can be achieved at 815 K for p-type material.

Tin selenide

In 2014, researchers at Northwestern University discovered that tin selenide (SnSe) has a ZT of 2.6 along the b axis of the unit cell. This is the highest value reported to date. This high ZT figure of merit has been attributed to an extremely low thermal conductivity found in the SnSe lattice. Specifically, SnSe demonstrated a lattice thermal conductivity of 0.23 W·m⁻¹·K⁻¹, which is much lower than previously reported values of 0.5 W·m⁻¹·K⁻¹ and greater. This SnSe material also exhibited a ZT of 2.3±0.3 along the c-axis and 0.8±0.2 along the a-axis. These excellent figures of merit were obtained by researchers working at elevated temperatures, specifically 923 K (650 °C). As shown by the figures below, SnSe performance metrics were found to significantly improve at higher temperatures; this is due to a structural change that is discussed below. Power factor, conductivity, and thermal conductivity all reach their optimal values at or above 750 K, and appear to plateau at higher temperatures. However, these reports have become controversial as reported in Nature because other groups have not been able to reproduce the reported bulk thermal conductivity data.

SnSe Performance Metrics

 

Although it exists at room temperature in an orthorhombic structure with space group Pnma, SnSe has been shown to undergo a transition to a structure with higher symmetry, space group Cmcm, at higher temperatures. This structure consists of Sn-Se planes that are stacked upwards in the a-direction, which accounts for the poor performance out-of-plane (along a-axis). Upon transitioning to the Cmcm structure, SnSe maintains its low thermal conductivity but exhibits higher carrier mobilities, leading to its excellent ZT value.

One particular impediment to further development of SnSe is that it has a relatively low carrier concentration: approximately 10¹⁷ cm⁻³. Further compounding this issue is the fact that SnSe has been reported to have low doping efficiency.

However, such single crystalline materials suffer from inability to make useful devices due to their brittleness as well as narrow range of temperatures, where ZT is reported to be high. Further, polycrystalline materials made out of these compounds by several investigators have not confirmed the high ZT of these materials.

Production methods

Production methods for these materials can be divided into powder and crystal growth based techniques. Powder based techniques offer excellent ability to control and maintain desired carrier distribution. In crystal growth techniques dopants are often mixed with melt, but diffusion from gaseous phase can also be used. In the zone melting techniques disks of different materials are stacked on top of others and then materials are mixed with each other when a traveling heater causes melting. In powder techniques, either different powders are mixed with a varying ratio before melting or they are in different layers as a stack before pressing and melting. 

There are applications, such as cooling of electronic circuits, where thin films are required. Therefore, thermoelectric materials can also be synthesized using physical vapor deposition techniques. Another reason to utilize these methods is to design these phases and provide guidance for bulk applications.

Significant improvement on 3D printing skills makes it possible for thermoelectric materials prepared from 3D printing inks, which are usually synthesized by dispersing inorganic powders to organic solvent or making a suspension. Thermoelectric products are made from special materials that absorb heat and create electricity. The requirement of having complex geometries that fit in tightly constrained spaces, makes 3D printing the ideal manufacturing technique. Also, printable materials that demonstrate good mechanical flexibility could be utilized for wearable thermoelectrics, which convert body energy to electricity.

Applications

Refrigeration

Thermoelectric materials can be used as refrigerators, called "thermoelectric coolers", or "Peltier coolers" after the Peltier effect that controls their operation. As a refrigeration technology, Peltier cooling is far less common than vapor-compression refrigeration. The main advantages of a Peltier cooler (compared to a vapor-compression refrigerator) are its lack of moving parts or refrigerant, and its small size and flexible shape (form factor). 

The main disadvantage of Peltier coolers is low efficiency. It is estimated that materials with ZT>3 (about 20–30% Carnot efficiency) would be required to replace traditional coolers in most applications. Today, Peltier coolers are only used in niche applications, especially small scale, where efficiency is not important.

Power generation

Thermoelectric efficiency depends on the figure of merit, ZT. There is no theoretical upper limit to ZT, and as ZT approaches infinity, the thermoelectric efficiency approaches the Carnot limit. However, no known thermoelectrics have a ZT greater than 3. As of 2010, thermoelectric generators serve application niches where efficiency and cost are less important than reliability, light weight, and small size.

Internal combustion engines capture 20–25% of the energy released during fuel combustion. Increasing the conversion rate can increase mileage and provide more electricity for on-board controls and creature comforts (stability controls, telematics, navigation systems, electronic braking, etc.) It may be possible to shift energy draw from the engine (in certain cases) to the electrical load in the car, e.g. electrical power steering or electrical coolant pump operation.

Cogeneration power plants use the heat produced during electricity generation for alternative purposes. Thermoelectrics may find applications in such systems or in solar thermal energy generation.