ааа Approaches for improving zT of thermoelectric materials Andrei Novitskii, Academic Research Center for Energy Efficiency, NUST MISIS Email: [email protected] @anovitzkij 1st Russian thermoelectric school September 10-12, 2021 ITMO Univesity sport center “Yagodnoe”, Russia

Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 2 What is the thermoelectric figure of merit? In 1909, the German engineer Edmund Altenkirch showed that the efficiency of a thermoelectric material is proportional to the square of the Seebeck coefficient, and the most efficient are those materials whose electrical and thermal conductivity ratio does not obey the Wiedemann-Franz law. He suggested the following figure of merit: 𝑧 = 𝛼2𝜎 𝜅 = 𝛼2 𝜅𝜌 here 𝑧 is the figure of merit (K–1), 𝛼 is the Seebeck coefficient, 𝜎 is the electrical conductivity, 𝜌 is the electrical resistivity, and 𝜅 is the total thermal conductivity, which represents the sum of the lattice and electronic contributions of the thermal conductivity 𝜅 = 𝜅𝑙𝑎𝑡 + 𝜅𝑒𝑙. In 1949 Ioffe proposed 𝑧𝑇 instead of 𝑧 as material’s thermoelectric figure of merit: 𝑧𝑇 = 𝛼2𝜎 𝜅 𝑇 = 𝛼2 𝜅𝜌 𝑇 A.F. Ioffe, Semiconductor Thermoelements, and Thermoelectric Cooling, Infosearch, London (1957). E. Altenkirch, Physikalische Zeitschrift 10, 560–580 (1909).

Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 3 What is the thermoelectric figure of merit? It is important to distinguish the material’s figure of merit 𝑧𝑇 and the device figure of merit 𝑍𝑇. For example, the maximum efficiency of the thermoelectric generator (device) can be expressed as 𝜂 = 𝑇ℎ𝑜𝑡 − 𝑇𝑐𝑜𝑙𝑑 𝑇ℎ𝑜𝑡 1 + 𝑍ത 𝑇 − 1 1 + 𝑍ത 𝑇 + Τ 𝑇𝑐𝑜𝑙𝑑 𝑇ℎ𝑜𝑡 where 𝜂 is the efficiency, 𝑇ℎ𝑜𝑡 and 𝑇𝑐𝑜𝑙𝑑 are the temperatures of the hot and cold junctions, respectively, ത 𝑇 is the average temperature between the cold and hot junctions. Here 𝑍 = 𝑧 only if temperature is assumed to be independent and thermoelectric properties of n- and p-type legs are matched, which is unrealistic approximation in majority of cases. For thermocouple consisting of n- and p-type legs: 𝑧ത 𝑇 = 𝛼𝑝−𝛼𝑛 2 ത 𝑇 𝜌𝑛𝜅𝑛+ 𝜌𝑝𝜅𝑝 2 while 𝑍ത 𝑇 = 𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑 1−𝜂 𝑇ℎ𝑜𝑡 1−𝜂 −𝑇ℎ𝑜𝑡 2 − 1 𝑧𝑇 ≠ 𝑍𝑇 G.J. Snyder, E.S. Toberer, Nat. Mater. 7 (2008) 105–114. G.J. Snyder, A.H. Snyder, Energy Environ. Sci. 10 (2017) 2280–2283.

Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 4 ZT calculator ZT calculator: http://thermoelectrics.matsci.northwestern.edu/thermoelectrics/ztcalc.html G.J. Snyder, A.H. Snyder, Energy Environ. Sci. 10 (2017) 2280–2283.

Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 5 TE module simulation from material’s properties https://tes.keri.re.kr/

Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 6 Strategies for the zT improvement 𝑧𝑇 = 𝛼2𝜎 𝜅 𝑇 = 𝛼2𝜎 𝜎𝐿𝑇 + 𝜅𝑙𝑎𝑡 𝑇 There are two most intuitive strategies that comes to mind when you see the formula for the figure of merit: (1) maximization of the power factor 𝛼2𝜎, (2) minimization of the lattice thermal conductivity 𝜅𝑙𝑎𝑡. However, all the transport properties 𝛼, 𝜎, and 𝜅 are strongly coupled and the main challenge is to tailor them in optimal way in order to develop thermoelectric material with superior performance. D.M. Rowe, CRC Handbook of Thermoelectrics: Macro to Nano, CRC Press, 2006.

H.J. Goldsmid, Sci. Technol. Adv. Mater. 22 (2021) 280–284. X. Zhang, et al., Sci. Adv. 6 (2020) eabc0726. A. Zevalkink, et al., Appl. Phys. Rev. 5 (2018) 021303. β Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 7 Quality factor So-called quality factor 𝐵 (or 𝛽) was firstly introduced in the late 50’s and it determines the maximum attainable 𝑧𝑇 for a given material into its most fundamental material properties, assuming optimized carrier concentration: 𝐵 ∝ 𝜇𝑤 𝜅𝑙𝑎𝑡 𝑇 Τ 5 2 here 𝜇𝑤 = 𝜇0 𝑚𝑑 ∗ 𝑚𝑒 Τ 3 2 is the weighted mobility, which is generally higher for multiband semiconductors. zTmax

H.J. Goldsmid, Sci. Technol. Adv. Mater. 22 (2021) 280–284. X. Zhang, et al., Sci. Adv. 6 (2020) eabc0726. A. Zevalkink, et al., Appl. Phys. Rev. 5 (2018) 021303. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 8 Quality factor Many researchers using 𝐵 as a metric for search of promising thermoelectric because 𝐵 depends on temperature, but not doping, while giving an insight of the potential thermoelectric performance of material for a given chemical potential. It also gives a valuable understanding on what kind of materials can potentially be effective as thermoelectric performance. For example, in the framework of deformation potential 𝜏0 ∝ Τ 1 𝑚𝑏 ∗ Τ 3 2, 𝜇0 = Τ 𝑒𝜏0 𝑚𝐼 ∗, and 𝑚𝑑 ∗ = 𝑁𝑣 Τ 2 3𝑚𝑏 ∗ , thus, 𝐵 ∝ Τ 𝑁𝑣 𝑚𝐼 ∗𝜅𝑙 . In this context, the most promising thermoelectrics should be among multiband semiconductors with low inertial effective mass.

J.R. Drabble, H.J. Goldsmid, Thermal Conduction in Semiconductors, Pergamon Press, (1963). D.T. Morelli, G.A. Slack, in: S.L. Shindé, J.S. Goela (Eds.), High Therm. Conduct. Mater., Springer, New York, 2006, pp. 37–68. Fig. from X. Qian, J. Zhou, G. Chen, Nat. Mater. (2021) G.A. Slack, J. Phys. Chem. Solids 34 (1973) 321–335. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 9 Lattice thermal conductivity Based on Leibfried and Schlömann model (lately corrected by Slack), the lattice thermal conductivity at high temperature 𝑇 > 𝜃𝐷 : 𝜅𝑙𝑎𝑡 = 𝐴 𝑘𝐵 𝜃𝐷 ℏ 3 ഥ 𝑀𝛿 𝛾2𝑁𝐴𝑣 𝑛 Τ 5 3𝑇 where 𝑁𝐴𝑣 is the Avogadro constant, 𝑛 is the number of atoms in the unit cell (molecule), 𝛿3 = 𝑉𝑎𝑡, 𝑉𝑎𝑡 is the average volume per atom, 𝐴 = 𝑓 𝛾 . Thus, several rules for low thermal conductivity can be formulated: (1) high mass of constituent atoms ( ഥ 𝑀𝛿𝜃𝐷 3 is maximized for light mass); (2) weak interatomic bonding; (3) complex crystal structure; (4) high anharmonicity. Conditions (1) and (2) means a low 𝜃𝐷, condition (3) means high 𝑛, and condition (4) means high 𝛾.

G. Slack in CRC Handbook of Thermoelectrics (ed. M. Rowe) p 407 – 440 (1995). Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 10 Phonon glass electron crystal In 1995 Glen A. Slack formulated the requirements for a thermoelectric materials in his concept of phonon glass electron crystal (PGEC) as follows: an effective thermoelectric material should conduct electricity effectively as a single crystal conductor and poorly conduct heat like glass. The most prominent representatives of this concept are skutterudites and clathrates. In such systems, the guest atom can “rattle” around the center in the cage. This rattling motion is expected to scatter the low-energy acoustic phonons and, thus, decrease the thermal conductivity to values typically seen in glass. The electrons, on the other hand, can flow through the network of the cage as in a crystalline metal. Crystal structure of RCoSb3 skutterudite (R is the rattling guest atom)

A.P. Novitskii, et al., Nanobiotechnology Reports 16 (2021) 294–307. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 11 Summary What are we looking for? Doped (optimal 𝑛) multiband semiconductors (high 𝜇𝑤) with complex crystal structure mainly composed of heavy elements (low 𝜅𝑙𝑎𝑡). Along with that we also should remember that the following things are also important: • The abundance of the constituent elements • Chemical and thermal stability of material under working conditions • Complexity of the synthesis technique and its scalability • Ability to develop paired leg of another type of conductivity

W.G. Zeier, et al., Nat. Rev. Mater. 1 (2016) 1–10. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 12 Approaches for improving zT Band convergence Carrier concentration optimization (Ioffe optimization) Seebeck coefficient increase Decrease of the thermal conductivity Quality factor optimization Increase of phonon scattering and lattice softening Complex crystal structures; heavy elements; high anharmonicity Materials that belong to phonon glass electron crystal (PGEC) concept Defect engineering Band structure engineering Compositing, texturation, modulation doping Resonant levels Spin-driven transport, magnetic doping, etc. 𝑧𝑇 = 𝛼2𝜎 𝜅 𝑇

H. Kim, et al., Phys. Status Solidi 254 (2017) 1600103. Y. Yu, et al., Mater. Today 32 (2020) 260–274. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 13 Phonon scattering 𝜅𝑙𝑎𝑡 = 1 3 න 𝐶 𝜔 𝜐2 𝜔 𝜏𝑐 𝜔 𝑑𝜔 Debye model: • phonon velocity is constant • scattering channels are independent of each other 1 𝜏𝑐 𝑥 = ෍ 𝑖 1 𝜏𝑖 𝑥 𝜅𝑙𝑎𝑡 = 𝑘𝐵 2𝜋2𝜐𝑚 𝑘𝐵 𝑇 ℏ 3 ∙ න 0 Τ 𝜃𝐷 𝑇 𝜏𝑐 𝑥 𝑥4𝑒𝑥 𝑒𝑥 − 1 2 𝑑𝑥

R. Hanus, et al., Adv. Mater. 31 (2019) 1900108 T.J. Slade, et al., Joule 5 (2021) 1168–1182. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 14 Lattice softening 𝜅𝑙𝑎𝑡 = 𝐴 𝜐𝑚 3 𝑇

A.F. Ioffe, Semiconductor Thermoelements, and Thermoelectric Cooling, Infosearch, London, 1957. H. Wang, et al., Adv. Funct. Mater. 23 (2013) 1586–1596. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 15 Carrier concentration optimization This approach is based on the theory of solid solutions proposed by A.F. Ioffe in 1956. Introduction of substitutional atoms leads to a distortion of the crystal lattice and thus to an increase in scattering of both phonons and charge carriers. This results in a decrease of the charge carrier mobility and the lattice thermal conductivity. However, their decrease is disproportional; accordingly, at some point in the chemical composition, a situation can be reached where the drop in the lattice thermal conductivity is much more significant than the drop in the mobility. In other words, it can be stated that this method is devoted to the optimization of the quality factor 𝐵 ∝ Τ 𝜇𝑤 𝜅𝑙𝑎𝑡 (see Figure). (PbTe)1–x (PbSe)x 𝐵800𝐾 : 𝐵300𝐾 = 1: 4

Y. Zheng, et al., Chem. Soc. Rev. 50 (2021) 9022–9054. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 16 Defect engineering

Y. Zheng, et al., Chem. Soc. Rev. 50 (2021) 9022–9054. D. Cheikh, et al., Joule 2 (2018) 698–709. Y. Liu, et al., J. Am. Chem. Soc. 133 (2011) 20112–20115. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 17 Defect engineering. Point defects • Dopants (charge carrier optimization) • Vacancies • Interstitial atoms • Antisites • Weak bonded guest atoms (rattlers) 1e per formula unit in R3 Te4 when xVR is formed the number of electrons is reduced by (1–3x) n-type

Y. Zheng, et al., Chem. Soc. Rev. 50 (2021) 9022–9054. D. Cheikh, et al., Joule 2 (2018) 698–709. Y. Liu, et al., J. Am. Chem. Soc. 133 (2011) 20112–20115. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 18 Defect engineering. Point defects • Dopants (charge carrier optimization) • Vacancies • Interstitial atoms • Antisites • Weak bonded guest atoms (rattlers) Stoichiometry control: vacancies at copper site leads to enhancement of the charge carriers concentration in BiCuSeO oxyselenides CuCu x → 𝑉Cu ′ + ℎ∙ p-type

S. Kim, et al., Science. 348 (2015) 109–114. Y. Sun, et al., J. Mater. Chem. C 9 (2021) 8506–8514. H.-S. Kim, et al., Mater. Horizons 3 (2016) 234–240. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 19 Defect engineering. Dislocations

S. Kim, et al., Science. 348 (2015) 109–114. Y. Sun, et al., J. Mater. Chem. C 9 (2021) 8506–8514. H.-S. Kim, et al., Mater. Horizons 3 (2016) 234–240. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 20 Defect engineering. Dislocations

M.G. Kanatzidis, Chem. Mater. 22 (2010) 648–659. J.F. Li, et al., NPG Asia Mater. 2 (2010) 152–158. Лекция «Введение в физику полупроводников» / 9 августа 2022 г. 21 Defect engineering. Nanostructuring 𝜅𝑙𝑎𝑡 = 𝑘𝐵 2𝜋2𝜐𝑚 𝑘𝐵 𝑇 ℏ 3 න 0 Τ 𝜃𝐷 𝑇 𝜏𝑐 𝑥 𝑥4𝑒𝑥 𝑒𝑥 − 1 2 𝑑𝑥 1 𝜏𝑐 𝑥 = 1 𝜏𝑔𝑏 𝑥 = 𝜐𝑚 𝐿𝐺 𝐵 ↑↑ 𝜇𝐻 ∝ 𝑒𝐿𝐺 8𝑚∗𝜋𝑘𝐵 𝑇− Τ 1 2𝑒− Τ Δ𝐸𝑏 𝑘𝐵𝑇

A.A. Usenko, et al., Scr. Mater. 96 (2015) 9–12. Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 22 Defect engineering. Nanostructuring

J. Li, et al., ACS Appl. Mater. Interfaces 11 (2019) 20064–20072. Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 23 Defect engineering. Nanostructuring

N. Wang, et al., npj Comput. Mater. 7 (2021) 18. Y. Pan, et al., Adv. Mater. 33 (2021) 2003168. A. Li, et al., Nat. Commun. 12 (2021) 5408. Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 24 Band structure engineering

N. Wang, et al., npj Comput. Mater. 7 (2021) 18. Y. Pan, et al., Adv. Mater. 33 (2021) 2003168. A. Li, et al., Nat. Commun. 12 (2021) 5408. Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 25 Band structure engineering

Y. Liu, et al, Appl. Phys. Lett. 106 (2015) 233903. Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 26 Band structure engineering

Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 27 Composites Filler type and properties: • Particles size • Particles shape • Uniform or anisotropic filling • Dimensionality • Conductivity type, including metal/semiconductor/insulator … Endless number of possible combinations and effects such as the effect of energy filtering of charge carriers, modulation doping, superparamagnetism, etc.

Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 28 Energy filtering effect Z. Ma, et al., Nanoscale 12 (2020) 1904–1911.

Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 29 Modulation doping Y.-L. Pei, et al., J. Am. Chem. Soc. 136 (2014) 13902–13908. BiCuSeO Bi1–x Bax CuSeO BiCuSeO + Bi1–x Bax CuSeO x = 0.25

Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 30 Superparamagnetism W. Zhao, et al., Nature 549 (2017) 247–251.

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Лекция «Введение в физику полупроводников» / 10 августа 2022 г. 32 Academic Research Center for Energy Efficiency Our website: http://energy.misis.ru/home_eng Introduction to Thermoelectricity guide: https://tinyurl.com/ateguide @anovitzkij @energy_misis [email protected]

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