Raveena
Gupta
^{ab},
Naveen
Kumar
^{a},
Prabhjot
Kaur
^{a} and
Chandan
Bera
*^{a}
^{a}Institute of Nano Science and Technology, Habitat Center, Phase-X, Mohali, Punjab-160062, India. E-mail: chandan@inst.ac.in
^{b}Centre for Nanoscience and Nanotechnology, Panjab University, Sector-25, Chandigarh-160036, India

Received
10th June 2020
, Accepted 27th July 2020

First published on 27th July 2020

The global energy crisis demands the search for new materials for efficient thermoelectric energy conversion. Theoretical predictive modelling with experiments can expedite the global search of novel and ecoconscious thermoelectric materials. The efficiency of thermoelectric materials depends upon the thermoelectric figure of merit (ZT). In this perspective, we discuss the theoretical model to calculate thermoelectric properties. Different scattering mechanisms of electrons and phonons are calculated using a simple model for the fast prediction of thermoelectric properties. Thermoelectric properties based on the simple model have shown more than 90% agreement with the experimental values. Possibility to optimize the figure of merit by alloying, defects, nanostructuring and band convergence is also discussed for layered chalcogenides of tin. In the case of doped materials, ion-impurity scattering is found to be dominating over electron–phonon scattering and the power factor can be optimized by tuning the former scattering rate. For phonon transport, alloy scattering is found to be the most dominating among all other scattering mechanisms. Theoretically, it is found that in the temperature range between 300 K and 800 K, SnSe_{0.70}S_{0.30} has the highest ZT with an efficiency of 17.20% with respect to Carnot efficiency. There could be 53.8% enhancement of the device efficiency in SnSe_{0.70}S_{0.30} compared to experimentally reported SnSe_{0.50}S_{0.50} in the medium temperature range (300 K to 800 K). Possible routes to achieve the best ZT in the medium temperature range are also discussed in this perspective.

The efficiency of a thermoelectric device is given by^{6}

(1) |

(2) |

(3) |

As the fabrication of technological products has to be cost-effective, the thermoelectric materials should be available at a low price, should have high stability with respect to temperature and give sufficient output. Therefore, for enhancing ZT, proper material selection is also necessary. Almost all metals and semiconductors exhibit thermoelectricity. The high range of carrier concentration in metals causes high electrical conductivity but reduces the Seebeck coefficient. Electronic thermal conductivity of the metals also increases with increasing temperature and therefore degrades the figure of merit in metals.

Hence, semiconductor materials are the most suitable thermoelectric materials^{9} for getting high ZT as they have high carrier mobility and a medium range of carrier concentration.

Though many thermoelectric materials have been developed by researchers, still research is going on to approach the ideal thermoelectric materials. Thermoelectric materials are classified as chalcogenides, clathrates, skutterudites, half-Heusler (HH) compounds, silicides, oxides, Zintl phase materials, etc. based on the structure and composition of the material.^{10} The bar graph in Fig. 1a shows ZT for various p-type and n-type binary thermoelectric materials studied theoretically and experimentally.^{11–26} Among these binary compounds, tin (Sn) based materials have promising potential for commercial applications due to their low cost, high stability and average high ZT in the medium temperature range.

Fig. 1 (a) Bar diagram showing the figure of merit of various binary p-type (green colour) and n-type (red colour) thermoelectric materials studied theoretically (striped) and experimentally (solid). The data have been taken from ref. 10–25. (b) Crystal structures of SnS, SnO and SnSe. |

Theoretically studied layered SnP_{3} has shown the highest ZT of 3.5 at 500 K.^{17} Experimentally, PbTe and SnSe at temperatures above 800 K have shown ZT > 2.^{15} Chalcogenides have seen to be good thermoelectric materials with the potential of improving ZT by doping, alloying and nanostructuring. A recent report by Zhao et al.^{27} showed that Cu_{2}Te-based compounds can be excellent thermoelectric materials if Cu deficiency is sufficiently suppressed by introducing Ag_{2} which increases carrier concentration and improves the ZT with a record-high value of 1.8, which is a 323% improvement over Cu_{2}Te. Though lead-based and copper-based chalcogenides show higher efficiency, tin-based chalcogenides are drawing more attention due to their non-toxicity. Tin-based chalcogenides are being continuously explored for higher figure of merit (ZT) values by proper optimization of their structural properties.^{15,23,28} Out of all chalcogenides of tin, SnO, SnSe and SnS are layered and exhibit promising properties which can be explored for obtaining higher efficiency. The thermoelectric properties of these compounds can be further optimized by alloying, doping or by preparing nanostructured layers by reducing the grain size of these materials.

In this perspective, we discuss the theoretical model used for the calculation of the thermoelectric properties and various optimization methods for improving the figure of merit. The thermoelectric properties of SnS, SnSe, SnO and their alloy compounds are investigated. First-principles investigation of structural, electronic, and phonon properties of undoped and doped compounds along with their alloys is also performed using rigid band approximation and alloys of tin chalcogenides with better ZT are predicted. The theoretical model discussed here can also be applicable for other layered and non-layered compounds.

The phase stable crystal structure of SnS and SnSe belongs to Pnma spacegroup while that of SnO belongs to P4/nmm spacegroup. The layered tetragonal structure of SnO with lone pair of electrons though not identical is similar to the layered orthorhombic structure of SnS and SnSe (Fig. 1b). The optimized lattice constants of SnS are a = 11.12 Å, b = 3.98 Å, c = 4.21 Å^{30} and that of SnSe are a = 11.49 Å, b = 4.17 Å, c = 4.29 Å^{30} which are in reasonable agreement with experimental reports by Chattopadhyay et al.^{31} and Wiedemeier et al.^{32} The bandgap of SnS and SnSe by GGA (Generalized Gradient Approximation) calculations is 0.62 eV and 0.30 eV respectively.^{30} The bandgap for both the materials improves on using the HSE06 functional^{30} with values 1.03 eV for SnS and 0.76 eV for SnSe, both in agreement with experimental values of 1.08 eV for SnS^{33} and 0.89 eV for SnSe.^{34}

For SnO also, a semi-local Generalized Gradient Approximation – Perdew–Burke–Ernzerhof (GGA-PBE) functional^{35,36} with a 8 × 4 × 4 k-mesh is used. The optimized cell parameters of SnO are determined to be a = 3.77 Å, b = 3.77 Å, and c = 4.66 Å, which are in reasonable agreement with an earlier experimental report by Moreno et al.^{37} (a = 3.80 Å, b = 3.80 Å, and c = 4.83 Å). The GGA + PBE calculations show no bandgap for SnO, though it has an optical bandgap of 0.68 eV reported experimentally by Miller et al.^{38} A Hubbard-U of 6 eV is employed as an onsite correction for band structure and density of states calculation. Inclusion of U results in a bandgap of 0.34 eV which is in agreement with the theoretical value reported by Miller et al.^{38} (0.32 eV).

(4) |

The scattering term of eqn (4) consists of non-linear integrals which are difficult to solve analytically; hence, numerical methods like Monte Carlo and finite element method and approximation methods like variational method and relaxation time approximation method are used to solve the transport distribution function. Some of the methods used for solving the Boltzmann transport equation are discussed briefly below.

(5) |

This method is widely used for studying the thermoelectric properties of materials. Jeng et al.^{42} presented a Monte Carlo simulation scheme to study the thermoelectric properties of nanocomposites with special attention paid to the implementation of periodic boundary conditions in Monte Carlo simulation. They applied the scheme to study the thermal conductivity of silicon germanium (Si-Ge) nanocomposites, which are of great interest for high efficiency thermoelectric material development. Gelmont et al.^{39} presented the results of an ensemble Monte Carlo simulation of the electron transport in gallium nitride (GaN) and showed that intervalley electron transfer plays a dominant role in GaN in high electric fields leading to a strongly inverted electron distribution and to a large negative differential conductance. Bera et al.^{43} also performed Monte Carlo simulation of thermal conductivity of Si nanowires to investigate the effect of phonon confinement on thermal transport.

Considering that the electron is under the effect of just the electric field and that the change in distribution function (f) from equilibrium distribution function (f_{0}) is very small, that is f ≈ f_{0}, the rate of change of distribution function is given as^{40}

(6) |

(7) |

(8) |

(9) |

X = Lϕ | (10) |

The transport coefficients obtained in this way are always less than their exact values as the results may not converge if the power series is terminated after a small number of terms.^{41} The inclusion of a large number of terms makes the computation very laborious. If expansion functions other than a simple power series are considered, it may give a rapid convergence.

Very few studies are there based on the variational method due to its limitations. A report by Casian et al.^{45} presented a systematic theoretical analysis of electronic states and thermoelectric transport in PbTe/Pb_{1−x}Eu_{x}Te quantum well structures by employing a more realistic well model than has been used up to now by using the variational method. The electrical conductivity, thermopower (Seebeck coefficient) and thermoelectric power factor as functions of the well width were studied for quantum well (QW) structures with (100) and (111) crystallographic orientations and different carrier densities and it was found that the power factor is greater in (100) QWs, but the more realistic the well model is, the lower is the power factor.

In this method the Boltzmann equation is reduced to the difference equation for the perturbation term and solutions are obtained by numerical iteration. The perturbation in the distribution function is determined by performing calculations for a number of energy points within the energy range of interest and at a particular iteration it is evaluated from the scattering-out term. The transport coefficients are hence evaluated using the perturbation. The contribution of the scattering-in term is not significant. A few iteration steps are usually required for a convergent result.

Rode utilized this method to calculate the drift mobility in several II–VI and III–V compounds.^{48,49} The thermoelectric powers of some III–V semiconductors were also calculated by him using this method.

(11) |

The motion of an electron is unhampered in a perfect crystal in which the wave function of the electron is given by stationary Bloch functions. The application of an external field in such crystals would uniformly accelerate the electron causing a linear increase of the drift velocity with time in the direction of the field.^{51} Such linear increase in drift velocity with time is not observed in real crystals. The average drift velocity of the electron reaches a limiting value which at low field is proportional to the magnitude of the applied field. The limit is set by the interaction of the electron with the imperfection of the crystal through the process referred to as the scattering or collision process.

The relaxation time approximation method is widely used for evaluating the thermoelectric properties of materials.^{52,53} Recently, Gupta et al.^{30} have shown that relaxation time approximation works very well for the calculation of thermoelectric properties of SnS, SnSe and SnSe_{1−x}S_{x} which are in good agreement with the experimental reports.^{54,55}

In the relaxation time approximation, the generalized transport coefficients are calculated from the transport distribution function given by^{56}

(12) |

In the above equation, v_{k} is the component of the group velocity of each carrier in the transport direction, E_{k} is the energy of that electronic state and τ_{k} is its total relaxation time which is calculated using Matthiessen's rule, i.e., where i runs over all scattering mechanisms. The moments of generalized transport coefficients are then given by^{56}

(13) |

The transport coefficients are obtained by applying boundary conditions. Identifying the conditions of zero temperature gradient and zero electric current, the transport coefficients are expressed as^{56}

σ = ζ^{(0)} | (14) |

(15) |

(16) |

Variation of the Seebeck coefficient (S) with carrier concentration (n) for layered chalcogenides (SnO, SnS and SnSe) is shown in Fig. 2. SnO shows the highest Seebeck coefficient compared to SnSe and SnS for both p-type and n-type compounds. For both p-type and n-type cases, the Seebeck coefficient decreases with an increase in carrier concentration as S ∝ n^{−2/3}. The p-type compounds have a Seebeck coefficient of lower magnitude compared to the n-type compounds as seen from the figure. The experimental values of the Seebeck coefficient^{38,54,55} are plotted in Fig. 2 for SnS, SnSe and SnO and correspond to 8.4 × 10^{16} cm^{−3}, 1.5 × 10^{17} cm^{−3} and 5 × 10^{17} cm^{−3} of carrier concentration respectively.

Fig. 2 Variation of the Seebeck coefficient with carrier concentration (measured in electrons per unit cell [e uc^{−1}]) for SnS, SnSe and SnO at 300 K along with experimental values^{38,54,55} (square – SnS, circle – SnO, triangle – SnSe). |

Fig. 3 Electron scattering mechanism adopted from ref. 41. |

4.1.1 Electron–electron scattering.
The effect of electron–electron scattering is more in degenerate semiconductors and can be neglected in non-degenerate semiconductors. The effect of this scattering process due to Coulomb forces was considered by Appel et al.^{58} in the framework of Kohler's variation principle. By considering the conduction electrons as a Fermi–Dirac gas of non-interacting free quasi-particles, each with charge e^{−} and mass m*, they took this scattering into account as a small perturbation to evaluate transport phenomena in non-polar semiconductors.

4.1.2 Piezoelectric scattering.
This scattering plays an important role and dominates deformation potential scattering at low temperatures in crystals having the piezoelectric effect. This scattering was first discussed by Meijer and Polder,^{59} who estimated the magnitude of the relaxation time for piezoelectric scattering in crystals with zinc blende symmetry. The piezoelectric mobility of electrons (or of holes) was also calculated by Zook^{60} for the three classes of known piezoelectric semiconductors, assuming energy surfaces which are ellipsoids of revolution with principal axes along the crystal axes. The piezoelectric scattering time is directly proportional to the square root of energy (E),

and the mobility is dependent on the square root of temperature (T),

τ_{piezo} ∝ E^{1/2} | (17) |

μ ∝ T^{1/2} | (18) |

4.1.3 Electron–phonon scattering.
Electron scattering by lattice vibration, i.e. by the phonon eigenstates, was described by Bardeen and Shockley.^{61} Phonons disturb the lattice, which moves the atoms from their original position in the lattice. An electron is affected by this position changing and may be liable to be deflected or scattered and hence the electron–phonon scattering rate is given by^{30}

where N(ε) is the electronic density of states, k_{B} is the Boltzmann constant, T_{L} is the lattice temperature, ρ is the density of the medium, u_{s} is the speed of sound and E_{D} is the band deformation potential which is defined as the change in the energy of an electronic level per unit of applied strain and is given by^{61,62}

where a is the lattice constant, a_{0} its equilibrium value, ∂(ε_{CBM/VBM} − ε_{core}) is the energy difference between the conduction band minima (CBM) or the valence band maxima (VBM) and a core level.

(19) |

(20) |

Electron–phonon relaxation time variation with energy (E − E_{F}) where E_{F} is the Fermi energy is shown in Fig. 4a. As the relaxation time is inversely proportional to the density of states of the system, the electron–phonon relaxation time is the lowest for SnSe and the highest for SnO. The e–ph relaxation time is calculated with a fixed phonon velocity. For SnO thermalization of electrons is much faster compared to SnS or SnSe. This scattering mechanism was used to calculate the electrical conductivity of SnO, SnS and SnSe (Fig. 4b). The electrical conductivity values of SnO lie in between those of SnS and SnSe making it a potential oxide thermoelectric and increase with an increase in carrier concentration. All p-type compounds show electrical conductivity of higher magnitude than n-type compounds as the scattering rate for p-type compounds is less compared to n-type compounds. For SnSe, it has electrical conductivity three times higher than that of SnS and SnO due to higher concentration (∼10^{18} cm^{−3}) of intrinsic charge carriers. It is a highly degenerate semiconductor and hence it shows quasi metallic features.

Fig. 4 (a) Variation of electron–phonon relaxation time with energy (E − E_{F}). (b) Variation of electrical conductivity with carrier concentration for pure SnO, SnSe and SnS at 300 K along with experimental values^{38,54,55} (square – SnS, circle – SnO, triangle – SnSe). |

4.1.4 Ionized-impurity scattering.
The ionized-impurity scattering rate comes into play when an impurity atom is added to an existing pure compound and it has a T^{3/2} partial mobility dependence,^{63}

with b defined as

(21) |

(22) |

In the above equation, n_{d} is the defect concentration, Z is the atomic number of the impurity element, m* is the effective mass of the carriers and n_{0} is the carrier concentration.

The ion-impurity scattering dominates electron–phonon scattering in all temperature ranges as shown in Fig. 5a for SnO. Gupta et al. had shown a hundred times increase in the electrical conductivity value for SnS and a five times increase for SnSe on silver doping. The n-type dopant iodine reduced the electrical conductivity of SnSe due to the pinning of Fermi energy inside the bandgap, making it a non-degenerate semiconductor. Silver (Ag) and indium (In), which are potential p-type dopants as their impurity states stabilize the Fermi level and enhance charge collection,^{38} along with gallium (Ga) are used to enhance the electrical conductivity of SnO. Maximum enhancement of the electrical conductivity (from 11.5 S m^{−1} to 55 S m^{−1}) for p-type SnO is observed for Ag doping (Fig. 5b). Carrier concentration after doping plays an important role in the electrical conductivity of tin chalcogenides and enhances their properties.

Fig. 5 (a) Variation of electron–phonon relaxation time and ion-impurity relaxation time with temperature. (b) Variation of electrical conductivity with carrier concentration for pure and doped SnO at 300 K along with experimental values^{38} (red circle – Ga doped SnO, black circle – pure SnO, blue circle – In doped SnO). |

4.1.5 Alloy scattering.
In the case of alloys, mobility is affected not only by impurities but also by disorder arising from atomic positions that break the lattice periodicity. The scattering rate for alloy scattering is given as^{64,65}

where Ω_{cell} is the unit cell volume, ħ is the reduced Planck constant and U is the alloy potential parameter whose value is considered to be 0.15 × 10^{5} similar to that considered for the SnSe_{1−x}S_{x} alloy.^{30}

(23) |

Alloy scattering dominates electron–phonon scattering; however it is lower compared to impurity scattering for doped materials (Fig. 6a). The electrical properties for the SnSe_{1−x}S_{x} alloy showed a similar behaviour as SnS and SnSe.^{30} The alloy SnSe_{0.70}S_{0.30} shows an increase in electrical conductivity from 0.9 S m^{−1} at 300 K to 100 S m^{−1} at 800 K upon increasing temperature. Though the electrical conductivity is increased for SnSe_{1−x}S_{x} compared to SnS, the values are inferior compared to that for SnSe at 300 K.^{30} The electrical properties for the SnO_{0.70}S_{0.30} alloy and SnO_{0.70}Se_{0.30} are shown in Fig. 6b. Similarly, the alloys of SnO also show an increase in electrical conductivity with temperature. We found a higher value of electrical conductivity of 30 S m^{−1} at 700 K for SnO_{0.70}Se_{0.30} compared to 15 S m^{−1} for pure SnO and 20 S m^{−1} for the SnO_{0.70}S_{0.30} alloy.

Fig. 6 (a) Variation of electron–phonon relaxation time, ion-impurity relaxation time and alloy relaxation time with temperature for SnO_{0.70}S_{0.30} (blue colour) and SnO_{0.70}Se_{0.30} (red colour) with indium doping. (b) Variation of electrical conductivity with temperature for pure SnO, SnO_{0.70}S_{0.30} and SnO_{0.70}Se_{0.30} along with experimental values^{38} for pure SnO. |

4.1.6 Intervalley scattering.
For the indirect band crystals, besides the above mentioned scattering, intervalley scattering which is the transition of electrons between states in different conduction band valleys is also important. This scattering can be formally treated in the same way as intravalley scattering by optical phonons.^{66} This scattering is included in a report by Bera et al.^{52} for the Si-Ge case by assuming that various indirect equivalent minima are isotropic and parabolic and is given as^{52}

where N is the total number of carriers, Z is the number of isotropic valleys at each band, D_{e} is the intervalley potential, f^{±} is the distribution function evaluated at energy δ′ = δ ± ħω_{p} and ħω_{p} is the phonon energy. The intervalley scattering rate is proportional to k′ and has a T^{−3/2} mobility dependence.

(24) |

The electronic transport properties and relaxation time for different scattering mechanisms can be also calculated using first-principles methods.^{67} Samsonidze et al.^{68} first principles calculations of the electron–phonon coupling demonstrated that the energy dependence of the electron relaxation time varies significantly with chemical composition and carrier concentration. Wang et al.^{69} combined the Boltzmann transport equation with an ab initio approach to compute the thermoelectric coefficients of semiconductors like silicon by taking electron–phonon, ionized impurity, and electron–plasmon scattering mechanisms into account. In another report^{70} the electrical conductivity accumulation with respect to electron mean free paths is compared to phonon thermal conductivity accumulation by using first-principles simulation and the thermoelectric properties of the bulk and nanostructured silicon are predicted. PERTURBO^{71} an open source software which mainly computes electron–phonon (e–ph) interactions and phonon limited transport properties in the framework of the Boltzmann transport equation (BTE) has recently been released to calculate the thermoelectric properties of materials using first-principles calculations and Wannier functions.

Within the relaxation time approximation, the lattice thermal conductivity is given by^{72}

(25) |

The relaxation time is calculated for different scattering mechanisms like Umklapp scattering and Normal scattering processes. Normal scattering becomes important in materials having a layered structure^{30} where anharmonic scattering overestimates the thermal conductivity.^{73} Alloy scattering and boundary scattering of phonons are also included in the calculations.

5.1.1 Umklapp (U) scattering.
The U-process is a phonon–phonon scattering process which is the most dominant process at high frequencies and high temperatures for crystals having less defects. The schematic of this process is shown in Fig. 7a where two large incoming phonons with wavevectors k_{1} and k_{2} result in an outgoing phonon with wave-vector k_{3}′ pointing outside the first Brillouin zone.

where p is an adjustable parameter given by

and V is the unit cell volume, s is the speed of sound, M is the average atomic mass and _{D} is the Debye temperature defined by

Hence, the net phonon momentum is not conserved in this process. The wave-vector is then mathematically transformed (k_{3}) to a point inside the first Brillouin zone. This transformation allows for scattering processes to physically occur. It is treated in the quasi-harmonic approximation which is based on the assumption that the harmonic approximation holds for every value of the lattice constant, which is viewed as an adjustable parameter. The scattering under this approximation is given by^{7,72,73}

(26) |

(27) |

(28) |

In the above equation, n is the number of atoms per unit cell and ^{2}; is the mode averaged-squared Grüneisen parameter given by

(29) |

The volume derivatives necessary to obtain the Grüneisen parameter are calculated for a specific volume range using the quasi-harmonic approximation. From the calculation of atomistic contribution of the Grüneisen parameter of SnO (Fig. 8a), SnS (Fig. 8b), and SnSe (Fig. 8c) by each of their constituent atoms, it is found that in SnO, the contribution of O atoms is higher in lower energy phonon modes while Sn contributes more in higher energy modes. In SnS, both atoms, Sn and S, contribute equally and the same is also seen for SnSe.

The Umklapp scattering model based on Grüneisen parameter calculations was also reported by Toher et al.^{74} and Toberer et al.^{75} for different structures. Miller and Toberer et al.^{76} used empirical fitting to calculate this parameter.

Inclusion of this scattering mechanism gives lattice thermal conductivity values near to experimental cases^{73} except for the case of layered Sn-chalcogenides where only inclusion of this scattering overestimates the thermal conductivity as reported by Gupta et al.^{30}

5.1.2 Normal (N) scattering.
The normal scattering plays an important role for layered chalcogenides along with the U-process. In the N-process, the net phonon momentum (G) is conserved. The schematic is given in Fig. 7b, where the sum of incoming phonon wave-vectors remains inside the first Brillouin zone.

where B_{N} is the normal phonon scattering rate coefficient, which for different a and b can be written as^{77}

The scattering rate is calculated as^{77}

(30) |

(31) |

The parameters a and b are adjustable and depend on the material's structure. The values are chosen to be a = 0.5 and b = 1.5 for thermal conductivity calculations of SnS, SnSe and SnO.

5.1.3 Defect scattering.
Scattering of phonons by point defects in crystals has been treated theoretically using perturbation theory and by self-consistent methods using Green function techniques by Klemens et al.^{78} These self-consistent methods are needed when the incident phonons are near or above an intrinsic resonance frequency of the defect. At sufficiently low frequencies, it is expected that perturbation theory will give good results if the unperturbed Hamiltonian of the system is chosen to minimize the perturbation. A defect or vacancy (V) gives rise to a mass difference equal to the mass of the missing atom, M_{v}. The breaking of bonds contributes in a way corresponding to a mass difference of 2M. This at low frequencies will cause a relaxation time given by^{79}

where

(32) |

(33) |

In the above equation V/n is the volume per atom and s is the speed of sound.

5.1.4 Alloy scattering.
Alloy scattering is incorporated into phonon transport by taking into account the mass disorder in the alloy.^{7} This scattering rate is calculated as

where v is the phonon velocity in the alloy, Γ is the scattering cross-section Γ = x(1 − x)(ΔM/M),^{2} ΔM = M_{A} − M_{B} and M = xM_{A} + (1 − x)M_{B}, M_{A} and M_{B} being the mass of alloy components A and B respectively. δ^{3} is the atomic volume of the alloy estimated as a weighted average,

where V_{A} and V_{B} are the atomic volumes of components A and B.

where A and B are the two components of the alloy and j stands for anharmonic or normal scattering.

(34) |

δ^{3} = xV_{A} + (1 − x)V_{B} | (35) |

For an alloy, the total anharmonic and normal scattering rate is calculated as

(36) |

5.1.5 Boundary scattering.
This scattering plays an important role for low-dimensional nanostructures and dominates over other scattering mechanisms. The boundary scattering is given by

where v is the phonon velocity, Λ is the characteristic length of the system and p represents the fraction of specularly scattered phonons. The parameter p can be computed using^{40}

(37) |

(38) |

In the above equation, η is the root-mean-square roughness of the surface and λ is the wavelength.

We use a diffusive model^{80} in Sn-based chalcogenide systems taking p = 0 and hence, the scattering rate reduces to

(39) |

The phonon modes in SnO have the maximum scattering rate 1.54 × 10^{12} s^{−1} compared to SnS and SnSe which leads to a maximum mean free path of 3.19 μm at 300 K. The relaxation time and the mean free path of the phonons decrease with temperature leading to a decrease of lattice thermal conductivity with temperature. Variation of thermal conductivity with characteristic length Λ for SnO, SnS and SnSe is shown in Fig. 9. SnO also shows a similar behaviour like SnS and SnSe^{30} with values varying from 1.78 to 1.60 as the characteristic length of the particle is reduced to 20 nm. Considering the diffusive type boundary scattering in these materials, the phonon mean free path is already very low, hence the sample sizes in the nanoscale have very little impact on thermal conductivity.

Fig. 10a shows the phonon band structure of SnO. Tin based chalcogenides have been reported to have thermal conductivity less than 5 W m^{−1} K^{−1}. Experimentally, SnTe^{28} has been seen to show the highest lattice thermal conductivity of 3.7 W m^{−1} K^{−1} compared to its counterparts at room temperature. SnSe has been reported to have an ultralow lattice thermal conductivity of 0.4 W m^{−1} K^{−1} at 300 K as shown by Zhao et al.^{15}

Fig. 10 (a) Phonon band structure of SnO. (b) Variation of lattice thermal conductivity, electronic thermal conductivity and total thermal conductivity with temperature of pure SnO. |

An experimental report^{55} showed SnSe to have a thermal conductivity of ∼1.27 W m^{−1} K^{−1} at 300 K. When only umklapp scattering is considered by the designed model, a very high value of 7.3 W m^{−1} K^{−1} is obtained. A similar observation is made in the case of SnS too.^{30} SnO has a thermal conductivity of 1.97 W m^{−1} K^{−1} as reported by Miller et al.

From our model, only U-processes overestimate the thermal conductivity values of layered chalcogenides. Therefore, by including the N-process as shown in Fig. 10b, the lattice thermal conductivity comes out to be well in agreement with the experimental value.

Fig. 11 shows the projected densities of states (PDOS) for each element in each structure SnS, SnSe and SnO. Starting from the low frequencies, the density of states is dominated by the massive Sn atom for all the three compounds SnS, SnSe and SnO. The contribution from the chalcogen atoms dominates in the higher frequency region. The Se atom has almost the same contribution as the Sn atom in SnSe while S and O atoms contribute less.

Fig. 11 Projected density of states, colored by respective elements, in each of the three structures (a) SnS (b) SnSe and (c) SnO. |

The lattice thermal conductivity can also be obtained by performing full ab initio calculations by using almaBTE^{83} or ShengBTE^{84} which are based on three-phonon processes using third order force constants. In a recent report,^{85} the lattice thermal conductivity of highly doped silicon was calculated considering the electron–phonon and defect scattering mechanisms both individually and together using first-principles. These ab initio techniques though accurate are very time consuming and based on a tedious calculation procedure. Fig. 12 shows the thermal conductivity obtained by different models compared with the experimental value. From these calculations, it is observed that inclusion of normal scattering processes gives results well in agreement with experimental values. It also shows that a simple model gives results comparable to the full ab initio based calculations in a much more faster way.

Fig. 12 Bar plot showing a comparison of the thermal conductivity of SnS, SnSe, and SnO using different models^{30,73} and experimental values.^{20,38,81,82} |

There are various ways in which the thermoelectric properties of thermoelectric materials can be improved to obtain higher efficiency of thermoelectric devices. Some of the ways of enhancing the ZT of Sn-based chalcogenides are discussed in the following sections.

Here we discuss a formalism that determines the defect concentration, doping levels and the impurity solubility in a material system. It includes first principles calculations of the formation energies and concentration of individual native defects as a function of the atomic chemical potentials of the host and dopant atoms as well as electron chemical potentials. Chemical potentials are preferred due to their direct relationship with the energies which are calculated from first principles and are subjected to some bounds that are related to the experimental growth conditions.^{86} The chemical potentials are allowed to vary over a restricted range determined by equilibrium thermodynamics. This defect calculation approach is based on three steps.^{86} At first the calculation of total energies of all types of defects (native as well as by some impurity) is done. Then the equilibrium concentrations and the resultant Fermi level using thermodynamic conditions like charge neutrality for all configurations are determined. This step depends on chemical potentials. Finally, the calculation of formation energy of the competing phases at that impurity concentration is done. By all the above mentioned calculations, the stable phase among all competing phases can be examined.

The total energies of defects and impurities are obtained from first principles calculations based on density functional theory (DFT)^{89} in local-density approximation (LDA) and ab initio pseudopotentials.^{90} These defect calculations are performed in a supercell geometry to provide adequate accuracy.

Formation energy for a native defect in the elemental system which is the difference between the calculated supercell energy and N-times the energy of a single bulk atom can be determined from N-atom defect supercell calculations.^{92} This analysis is not applicable for a compound semiconductor as the energies and concentration of the native defects and the single atom also depend on the environment. For such cases, the defect formation energy can be defined by introducing an external reservoir of dopant atoms. The dopant atoms may be added to the crystal from the reservoir or vice versa. In this case, the energy of the atom is constant in the reservoir and is in thermal equilibrium with the crystal which allows one to determine a single energy value of the atom.^{90} This allows determination of the formation energy of any defect. The defect formation energy with respect to the standard state of the defect can be written as^{91}

(40) |

(41) |

μ_{Sn} ≤ 0 | (42) |

μ_{X} ≤ 0 | (43) |

μ_{Sn} + 2μ_{X} ≤ 3ΔH_{f}(SnX_{2}) | (44) |

For maintaining stability in the crystal,

μ_{Sn} + μ_{X} = 2ΔH_{f}(SnX) | (45) |

The crystal structure after Ag doping and the relevant chemical potential limits for doping SnS are shown in Fig. 13a and b respectively. The calculated defect formation energy as a function of μ_{e} for all these defects in Sn-rich and S-rich limits is shown in Fig. 14. It was found that not only in the S-rich limit the unfavourable V_{Sn} destabilizes, but also the Ag_{Sn} defect stabilizes. In ref. 91 defect formation energy calculation for SnSe is discussed and in ref. 38 stable defect formation in SnO is discussed.

Fig. 13 (a) The crystal structure of SnS showing Ag substituted on the Sn site in the SnS lattice. (b) Calculated chemical potential limits for Ag doped SnS. The shaded area is for the allowed equilibrium growth conditions. Reprint from ref. 91. |

Fig. 14 Defect formation energies for Sn-rich and S-rich limits in SnS as a function of the electron chemical potential. The slope of the lines corresponds to the charge of defects. The solid lines and dashed lines correspond to defects at the Sn and S position respectively. Reprint from ref. 91. |

The equilibrium defect concentration can be determined from the calculated formation energies as

(46) |

(47) |

(48) |

(49) |

In the above equations, E_{CBM} is the energy of the conduction band minimum, n(ε) is the electronic density of states of the defect free crystal and f(ε,μ_{e}) is the Fermi distribution. Fig. 15 shows the calculated carrier concentration based on defect formation energies in SnS and SnSe, which indicates that intrinsic carrier concentration in the low-temperature phase of SnSe (∼10^{18} cm^{−3}) is two orders of magnitude higher than in SnS (∼10^{16} cm^{−3}) and carrier concentration in the low temperature phase (Pnma structure) of SnSe can still be further optimized by silver doping.^{91}

Fig. 15 Carrier concentration due to defects in SnS as a function of fabrication temperature. All lines correspond to the S-rich limit except for phosphor doping, which is in the Sn-rich limit. Also shown in thin lines are the calculated carrier concentrations in the low temperature phase of SnSe. Reprint from ref. 91. |

Distortion of the electronic density of states (DOS) is also a potent mechanism to increase the thermopower of thermoelectric semiconductors. One complementary band-structure engineering approach that promises to enhance the thermoelectric power is the use of ‘resonant impurities’ as dopants. The review by Nemov^{94} showed that resonant impurities are a concept that was introduced in solid state physics for metals first, and they also exist in many semiconductors. L. A. Falk’ovskii pointed out on a theoretical basis that resonant levels (RL) are likely to exist in very narrow-gap semiconductors, particularly those with a strongly non-parabolic dispersion relation.^{95}

The narrower the gap, the higher the probability that an impurity level will coincide with a band. RLs involve a coupling between electrons of the impurity and electrons in the conduction or the valence band of the host solid. The impurity atoms have multiple electron energy levels corresponding to either a bound state or an extended state. Resonant donor impurities would have electronic energy levels for which E_{D} (equivalent to the thermal excitation energy of dopants in conventional semiconductors) is negative or positive (depends on donor or acceptor type), with the impurity level falling inside the conduction band or valence band and coinciding with energies of extended states.^{96} This state is called a resonant state. As this state has now the same energy as an extended state, the two will resonate to build up two extended states of slightly different energies; these in turn will have the same energies as other extended states with which they will resonate in turn, and so on. Consequently, the resonant state develops a certain width Γ. Both E_{D} and Γ are essential design parameters in optimizing ZT. Mahan and Sofo showed that, the narrower the Γ, the higher the ZT.^{97} The resonant level enhances the thermoelectric properties due to its excess density of states at their corresponding energy levels and the resonant state diffuses conduction electrons in a way that is extremely sensitive to their energy known as resonant scattering. The chalcogens S, Se, and Te are mentioned as potential resonant donors in heavy compounds like InAs and InSb.^{97} A significantly enhanced Seebeck coefficient of 116 μV K^{−1} at 300 K was observed by codoping of Mn and In in SnTe due to a hump in DOS due to resonance levels.^{98} Al doping in PbSe created resonant states in the conduction band resulting in an increase of the local density of states near the Fermi level causing the Seebeck coefficient to be about 40% higher than that of the Cl-doped PbSe sample without resonant states.^{99} These studies suggest that resonant impurity levels can be used to maximize the thermoelectric figure of merit, by maximizing the thermoelectric power at a given carrier concentration.

Doping in SnX (X = S, Se, O and Te) is discussed here to understand the importance of doping and defect engineering. Na doped SnS also has very good thermoelectric properties and has ZT = 0.65 due to its increased carrier concentration (10^{19} cm^{−3}).^{100} Further, a very high ZT = 1.6 was reported for substituting S with Se by Wenke et al.^{23} A first principles study of Bi doped SnS showed that the ZT value increased from 0.16 to 0.36 at 1.56% Bi concentration.^{101} The study of In (5% and 15%) doped SnS samples and Sb (5% and 15%) doped SnS samples showed an increase in electrical conductivity and Seebeck coefficient with temperature indicating p-type semiconducting behaviour of all samples.^{102}

A high figure of merit (ZT) = 1.1 at 773 K was reported in n-type Sb doped SnSe microplates due to high carrier concentration (3.94 × 10^{19} cm^{−3})^{103} and ZT = 2.2 was obtained at 733 K for Bi doped n-type SnSe crystals having a carrier density of 2.1 × 10^{19} cm^{−3}.^{104} An experimental study also showed that for 2 wt% Cu doped SnSe, a ZT value of 0.7 ± 0.02 can be obtained at 773 K.^{105}

It was found by Tan et al. that Bi and In co-doping in SnTe can enhance the Seebeck coefficient, resulting in a significant increase in the ZT value to 1.26 at 900 K.^{106} Mn, Bi and Sb co-doped SnTe has low lattice thermal conductivity and ZT ∼ 1 at 773 K^{107} indicating it to be a good thermoelectric material. All of the above mentioned studies suggest that SnS, SnSe, SnO and SnTe are good thermoelectric materials and can be used to design and optimize the advanced thermoelectric devices.

Our theoretical calculations based on relaxation time approximation and the rigid band model show that extrinsic doping largely affects the ion-impurity scattering rate. In Sn-based chalcogenides, electrical properties can be improved via extrinsic doping. Theoretical studies show ∼10^{2} times improvement for p-type doping and ∼10 times improvement for n-type doping in SnSe_{0.70}S_{0.30}.^{30}

Substitutional alloying has a remarkable effect on the thermoelectric behaviour of the material as it produces different kinds of defect atoms on lattice sites changing volume and hence thermoelectric properties.^{108} Formation of an alloy is conditioned to the structure of components forming the alloys. The crystal structure of the forming materials should be approximately similar. This is generally achieved by choosing materials with iso-electronic elements as they tend to make a similar kind of structure.

Among chalcogenides of tin (Sn), SnS has orthorhombic structure with space group Pnma and is being continuously investigated as a thermoelectric material.^{109} SnSe also has Pnma structure.^{110} Wang et al.^{111} showed that in the case of undoped SnS, which possesses low carrier concentration and inferior TE properties, its thermoelectric performance can be improved by Se alloying. They showed that narrowing of the band gap and flattening of the valence band shape contribute to excellent electrical transport properties, resulting in a maximum power factor of ∼6.0 mW cm^{−1} K^{−2} and a record high ZT of 0.70 at 873 K. In another report^{112} the thermal conductivity of SnSe was greatly reduced upon Te substitution (SnSe_{1−x}Te_{x}) due to alloy scattering of phonons as well explained by the Debye model. Due to the increased carrier concentration by Na-doping in this alloy compound, the thermoelectric figure of merit (ZT) was enhanced in the whole temperature range with a maximum value of 0.72 obtained at a relatively low temperature (773 K) for Sn_{0.99}Na_{0.01}Se_{0.84}Te_{0.16}.

In another Sn based non-layered chalcogenide, SnTe alloying was found to be very important for optimizing TE properties. Zhang et al.^{14} reported that both the power factor and thermal conductivity of SnTe can be simultaneously improved by introducing its analogues (PbTe, PbSe, PbS, SnSe, and SnS) into the SnTe matrix. They found that the power factor of SnTe is greatly enhanced as alloying with its analogues modifies its band structure. They showed that Seebeck coefficients at room temperature show an increasing trend with increasing PbTe fractions. The Seebeck coefficient increases from ∼ 170 μV K^{−1} for Sn_{0.97}Bi_{0.03}Te to ∼200 μV K^{−1} for Sn_{0.97}Bi_{0.03}Te with 3 wt% PbTe at 900 K. This increase is the result of convergence of the two valence bands, the light valence band (L) and the heavy valence band (Σ), which facilitates charge carrier injection and the participation of the heavy hole band in the carrier transport leads to increment in power factor values and hence higher figure of merit. An increase in electrical conductivity is also seen for SnO when it is alloyed with SnSe and SnS as seen in Fig. 7b.

Alloying also plays a major role in reducing the lattice thermal conductivity. It affects a particular range of phonons and scatters them selectively, thus enhancing the thermoelectric figure of merit. In the literature, the thermal conductivity of binary compounds SnS and SnSe is reduced when alloy SnSe_{1−x}S_{x} is formed. Gupta et al.^{30} show that the thermal conductivity of SnSe_{0.70}S_{0.30} is 1.3 times lower than that of SnSe and ∼2 times lower than that of SnS. This reduction in thermal conductivity for the alloy case is mainly due to atomic mass difference, which increases scattering and hence impedes heat transport by short wavelength phonons. The lattice thermal conductivity also decreased in SnO alloys SnO_{0.70}Se_{0.30} and SnO_{0.70}S_{0.30} giving higher ZT compared to pure SnO as seen in Fig. 17. ZT for SnO_{0.70}S_{0.30} is higher at 300 K but is the same at 800 K. SnO and SnSe alloying enhances the figure of merit twice from 0.0022 to 0.0044 at 800 K.

Fig. 17 Variation of figure of merit with temperature for SnO, SnO_{0.70}Se_{0.30} and SnO_{0.70}S_{0.30} with experimental values for pure SnO.^{38} |

Thus, alloying can enhance the thermoelectric properties of materials. It leads to modification of electronic band structure by shifting of light and heavy valence bands leading to band convergence which is discussed in the next subsection.

Band convergence can significantly enhance the Seebeck coefficient without detrimental effects on the electrical conductivity in SnTe.^{14} Acharya et al.^{28} also showed that with heavy atomic mass and strong spin–orbit coupling, even the mild doping of Yb (∼5%) is enough to create a degeneracy via band-convergence which enhances the density of states near the Fermi level and improves the overall thermoelectric response and is equivalent to 9% of Mg doping. The supercell approach is used for band convergence study in SnTe^{28} and PbTe.^{113} There is no report on band convergence for SnS, SnSe and SnO to date.

In 1990, Hicks, Dresselhaus and Harman suggested^{117,118} that the figure of merit could be improved if electrons were confined in two dimensions using quantum well superlattices, where superlattices are multilayers of thin films of the order of several nanometers in thickness. Hicks and Dresselhaus later extended their work to include one-dimensional conductors such as nanowires.^{119} The primary reason for the enhancement in the figure of merit in low-dimensional structures is due to an increase in the electronic density of states per unit volume, which leads to an improved thermopower. Boltzmann transport equations are also applied at lower dimensions and their applicability is justified by considering carrier-boundary scattering.

Liu et al.^{120} recently reported SnSe/SnS hetero-nanosheet tuning by the epitaxial growth of SnSe on the few layers of SnS nanosheets. The heterojunction nano-interface optimizes the carrier/phonon transport behavior by the energy filtering effect leading to an increase in power factor from 2.2 μW cm^{−1} K^{−2} to 3.21 μW cm^{−1} K^{−2} at 773 K compared to pristine SnSe and a significant reduction in thermal conductivity from 0.65 W m^{−1} K^{−1} to 0.48 W m^{−1} K^{−1} at 773 K. They reported a maximum ZT of 0.5 at 773 K in the SnSe/SnS hetero-nanosheets, which is 89% higher than that of pristine SnSe. Hence, their approach proved to be a promising strategy to design high performance thermoelectric materials.

A recent report^{121} show the development of a new solution synthesizing method (in situ magnetic field-assisted hydrothermal synthesis) for achieving new nanostructured SnSe integrated with Se quantum dots. This method leads to reduction in critical nucleation energy and enhancement of the nucleation during the hydrothermal synthesis process as a high magnetic field is applied, leading to the presence of the homogeneous distribution of Se quantum dots and smaller nanograins. SnSe nanostructures and Se quantum dots (Se quantum dot/Sn_{0.99}Pb_{0.01}Se nanocomposite) enhance the density of states and cause the energy filtering effect which contribute towards a significant enhancement in the Seebeck coefficient and power factor which further leads to a high figure of merit (ZT) of ∼2.0 at 873 K.

Xu et al.^{122} adopted a strategy of interface engineering to design oriented nanopillar structure in SnTe films via the one-step thermal evaporation method. The analysis of the electrical and thermal properties illustrates that the SnTe film with highly oriented growth shows enhanced thermoelectric performance. The excellent properties are mainly attributed to the special nanopillar structure, which may facilitate electrons' transport while significantly scattering the phonons. A maximum power factor of 19.8 μW cm^{−1} K^{−2} and a thermal conductivity of 3.54 W m^{−1} K^{−1} are obtained making the work a promising approach to enhance the thermoelectric performance of SnTe films.

Considering diffusive type surface scattering no reduction in the thermal conductivity of layered tin based chalcogenides has been observed as seen in Fig. 19. Gupta et al.^{30} showed that a reduction of diameter to 100 nm changes the thermal conductivity of SnSe_{0.70}S_{0.30} by only ∼10%, therefore not affecting the figure of merit of the materials. SnO_{0.70}S_{0.30} and SnO_{0.70}S_{0.30} also show the same behaviour on nanostructuring.

Fig. 19 Bar plot showing a comparison of theoretically calculated figure of merit^{30} and experimental values for tin-based chalcogenides.^{30,38,54,55,125} |

As the electron generally has a smaller wavelength compared to the phonon, nanostructuring will not affect electron scattering much in Sn-based chalcogenides. Electron and phonon dispersions can also change in nanostructure systems but full ab initio calculations based on DFT will be very expensive to study the nanostructure electron and phonon dispersions. The tight binding model can be suitable for this type of calculation. So far there is no report on the electronic and phonon band structure of nanowires or nanoparticles based on Sn-based chalcogenides. Recently a few studies have reported thermoelectric properties of monolayered Sn based chalcogenide compounds.^{123,124}

The theoretical model discussed in this perspective gives results well in agreement with experimental values.^{38,54,55,125} SnO has the lowest ZT among all other chalcogenides. The figure of merit improves on alloying with the highest value of 1.3 obtained theoretically for SnSe_{0.70}S_{0.30} which is not reported yet experimentally. From theoretical calculations, it is observed that SnSe_{0.70}S_{0.30} will be the best Sn-based thermoelectric material.

When the device efficiency is calculated from eqn (1), large enhancement is also observed in SnSe_{0.70}S_{0.30}. The variation of η/η_{0} with temperature for ideal ZT is shown in Fig. 20. The different lines in the graph are calculated by taking Z as constant for the entire temperature range. The scatterers represent the experimental and theoretical data for layered tin-based chalcogenides and their alloys.^{30,38,54,55,125}η/η_{0} for these points was calculated by taking average of ZT values at T_{C} = 300 K and T_{H} = 800 K. It is seen that the alloy SnSe_{0.70}S_{0.30} has the maximum efficiency of 17.20% with respect to Carnot efficiency at 800 K.

Fig. 20 Variation of η/η_{0} with temperature for tin-based chalcogenides with theoretically calculated and experimental values.^{30,38,54,55,125} |

This is a 53.8% enhancement in the efficiency value from the experimentally reported case of SnSe_{0.50}S_{0.50}. From this plot it is seen that thermoelectric device efficiency with respect to Carnot efficiency can be improved in the medium temperature range by alloying and proper extrinsic doping.

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