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[3/6] Mobility, quality factor, zT

[3/6] Mobility, quality factor, zT

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Andrei Novitskii

March 30, 2021
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  1. ааа Charge carrier mobility, quality factor and thermoelectric efficiency Andrei

    Novitskii, Academic Research Center for Energy Efficiency, NUST MISIS Email: novitskiy@misis.ru @anovitzkij
  2. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    2 Charge carrier (drift) mobility Let’s remember what is the carrier mobility?[1] 𝑣𝑑 = 𝑒𝜏 𝑚𝐼 ∗ 𝐸 = 𝜇𝑑 𝐸 and 𝜎 = 𝑒𝑛𝜇𝑑 Why is that so important? scattering
  3. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    3 Charge carrier mobility In materials with multiple scattering mechanisms affecting the carrier mean free path, the temperature dependence of charge carrier mobility is complex and total 𝜇 can be evaluated by Matthiessen’s rule,[2,3] which assumes scattering channels are independent of each other 𝜇−1 = ෍ 𝑖 𝜇𝑖 −1 where 𝜇𝑖 represents the mobility of a specified scattering mechanism. The dominant scattering mechanism can be determined using log-log plot from the temperature dependence of the Hall mobility fitted by a power-law 𝜇 ∝ 𝑇𝑚, where 𝑚 is the scattering factor of the carriers.
  4. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    4 Electrical conductivity 𝜎 𝑇 = 𝑒𝑛(𝑇)𝜇(𝑇) For semiconductors: 𝑛 𝑇 = 𝑁𝑐 𝑒−𝐸𝑐−𝐹 𝑘𝐵𝑇 , 𝜇(𝑇) ∝ 𝑇𝑚 For metals: 𝑛 = 1 3𝜋2 2𝑚0𝐸 ℏ2 Τ 3 2 , 𝜇(𝑇) ∝ 𝑇𝑚
  5. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    5 Charge carrier mobility 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 𝐹𝑗 𝜂 = න 0 ∞ 𝜀𝑗 1 + 𝑒𝜀−𝜂 𝑑𝜀 here 𝜇0 is typically called as the carrier concentration independent mobility parameter or free mobility parameter. 𝜇0 value can be obtained through the analysis of the Hall mobility data. Degenerate limit: 𝜇𝐻 ∝ 𝜇0 𝑚∗ 2𝑇𝑛 Τ 1 3 Non-degenerate limit: 𝜇𝐻 = 𝜋 2 𝜇0 For a single parabolic band semiconductor, the Hall mobility can be represented by
  6. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    6 Scattering mechanisms Scattering mechanisms Defect scattering Crystal defects Impurity Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intervalley Acoustic Optic Intervalley Acoustic Deformation potential Piezoelectric Optic Polar Nonpolar
  7. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    7 Scattering mechanisms Scattering mechanisms Defect scattering Crystal defects Impurity Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intervalley Acoustic Optic Intervalley Acoustic Deformation potential Piezoelectric Optic Polar Nonpolar
  8. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    8 Acoustic phonon scattering 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 For acoustic phonon scattering: 𝜇0 = 𝑒𝜋ℏ4𝐶𝑙𝑙 2 𝑘𝐵 𝑇 Τ 3 2 𝑚𝑏 ∗ Τ 3 2 𝑚𝐼 ∗Δ𝑑𝑒𝑓 2 where ℏ is the reduced Planck's constant, 𝐶𝑙𝑙 is the elastic constant for longitudinal vibrations (𝐶𝑙𝑙 = 𝑑𝑣𝑙 2, where 𝑑 is the density, 𝑣𝑙 is the longitudinal component of sound velocity), 𝑚𝑏 ∗ is the effective mass of a single valley, 𝑚𝐼 ∗ is the inertial effective mass (for the isotropic spherical case 𝑚𝑏 ∗ = 𝑚𝐼 ∗), and Δ𝑑𝑒𝑓 is the deformation potential characterizing the carrier-phonon interaction.[2,4] It should be mentioned, that 𝐶𝑙𝑙, 𝑚∗ and Δ𝑑𝑒𝑓 can also depend on temperature.
  9. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    9 Scattering mechanisms Scattering mechanisms Defect scattering Crystal defects Impurity Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intervalley Acoustic Optic Intervalley Acoustic Deformation potential Piezoelectric Optic Polar Nonpolar
  10. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    10 Ionized impurity scattering 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 For ionized impurity scattering: 𝜇0 = 8 2𝑘𝐵 Τ 3 2 𝜋 Τ 3 2𝑒3 𝜀𝜀0 2 𝑁𝐼 𝑚𝑑 ∗ Τ 1 2 𝑍2 𝑇 Τ 3 2 ln 1 + 3𝜀𝜀0 𝑘𝐵 𝑇 𝑁 𝐼 Τ 1 3𝑍𝑒 2 This expression for the mobility encompasses the Conwell-Weisskopf and Brooks-Herring.[2,4] It is sufficient when 𝑇2 ≪ 𝑁𝐼 Τ 2 3𝑍2𝑒4 9 𝜀𝜀0 2𝑘𝐵 2
  11. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    11 Scattering mechanisms Scattering mechanisms Defect scattering Crystal defects Impurity Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intervalley Acoustic Optic Intervalley Acoustic Deformation potential Piezoelectric Optic Polar Nonpolar
  12. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    12 Alloy disorder scattering 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 For alloy disorder scattering: 𝜇0 = 16𝑒ℏ4 9 2𝑧(1 − 𝑧) 𝑘𝐵 𝑇 Τ 1 2 𝑁0 𝑚𝑑 ∗ Τ 5 2 𝑈2 where 𝑧 is fractional concentration of the solid solution and 𝑁0 is the number of atom per unit volume. The potential energy fluctuation caused by alloy disorder is characterized by 𝑈, which is analogous the deformation potential Δ𝑑𝑒𝑓.[2,5]
  13. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    13 Scattering mechanisms Scattering mechanisms Defect scattering Crystal defects Impurity Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intervalley Acoustic Optic Intervalley Acoustic Deformation potential Piezoelectric Optic Polar Nonpolar
  14. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    14 Piezoelectric potential scattering 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 For piezoelectric potential scattering: 𝜇0 = 16 2𝜋 3 ℏ𝜀𝜀0 𝑚∗ Τ 3 2𝑒𝐾2 𝑘𝐵 𝑇 − Τ 1 2 with 𝐾 = ൘ 𝑒𝑝 2 𝐶𝑙𝑙 𝜀𝜀0 + ൘ 𝑒𝑝 2 𝐶𝑙𝑙 𝑒𝑝 being the piezoelectric coefficient. In strongly ionic crystals, e.g. II–VI semiconductors, the piezoelectric scattering can be stronger than the deformation potential scattering.[2]
  15. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    15 Scattering mechanisms Scattering mechanisms Defect scattering Crystal defects Impurity Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intervalley Acoustic Optic Intervalley Acoustic Deformation potential Piezoelectric Optic Polar Nonpolar
  16. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    16 Polar optical scattering 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 For polar optical scattering (where 𝑇 ≪ 𝜃𝐷): 𝜇0 = 𝑒 2𝑚∗𝛼𝑝 𝜔0 𝑒 Τ 𝜃𝐷 𝑇 where in the scattering mechanism the absorbed or emitted phonon energy ℏ𝜔0 is comparable to the thermal energy of the carriers,[2] 𝛼𝑝 is the dimensionless polar constant: 𝛼𝑝 = 1 137 𝑚∗𝑐2 2𝑘𝐵 𝜃𝐷 1 𝜀 ∞ − 1 𝜀 0
  17. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    17 Scattering mechanisms Scattering mechanisms Defect scattering Crystal defects Impurity Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intervalley Acoustic Optic Intervalley Acoustic Deformation potential Piezoelectric Optic Polar Nonpolar
  18. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    18 Dislocation scattering 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 For dislocation scattering (for n-type semiconductor): 𝜇0 = 30 2𝜋 𝜀𝜀0 2𝑑2 𝑁𝑑𝑖𝑠𝑙 𝑒3𝑓2𝐿𝐷 𝑚∗ Τ 1 2 𝑘𝐵 𝑇 Τ 3 2 𝑑 being the average distance of acceptor centers along the dislocation line, 𝑓 their occupation rate, 𝑁𝑑𝑖𝑠𝑙 the area density of dislocations and 𝐿𝐷 = Τ 𝜀𝑘𝐵 𝑇 𝑒2𝑛 the Debye screening length. Dislocations can contain charge centers and thus act as scattering centers. The deformation has introduced acceptor-type defects reducing the mobility in particular at low temperatures (similar to ionized impurity scattering).[2]
  19. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    19 Grain boundaries scattering 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 For grain boundaries scattering: 𝜇0 = 𝑒𝐿𝐺 8𝑚∗𝜋𝑘𝐵 𝑇− Τ 1 2𝑒− Τ Δ𝐸𝑏 𝑘𝐵𝑇 where 𝐿𝐺 is the grain size. Grain boundaries contain electronic traps whose filling depends on the doping of the bulk of the grains. Charges will be trapped in the grain boundaries and a depletion layer will be created.[2,4] At low doping, the grains are fully depleted and all free carriers are trapped in the grain boundaries. This means low conductivity, however, no electronic barrier to transport exists. At intermediate doping, traps are partially filled and the partial depletion of the grain leads to the creation of an electronic barrier Δ𝐸𝑏 hindering transport since it must be overcome via thermionic emission. At high doping the traps are completely filled and the barrier vanishes again.[4]
  20. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    20 Scattering mechanisms Scattering mechanisms Defect scattering Crystal defects Impurity Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intervalley Acoustic Optic Intervalley Acoustic Deformation potential Piezoelectric Optic Polar Nonpolar m = –3/2 m = 1/2 m = 3/2 m = 0 mGB = –1/2 mdisl = 3/2 m = –1/2 m = –1/2
  21. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    21 Further reading
  22. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    22 Charge carrier mobility 𝜇0 −1 = ෍ 𝑖 𝜇0,𝑖 −1 = 1 𝜇0,𝑝ℎ + 1 𝜇0,𝑖𝑜𝑛 + 1 𝜇0,𝑔𝑏 + ⋯ 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 𝑛 𝜂 = 4𝜋 2𝑚𝑑 ∗ 𝑘𝐵 𝑇 ℎ2 Τ 3 2 𝐹𝑟+1 𝜂
  23. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    23 Charge carrier mobility 𝜇0 −1 = ෍ 𝑖 𝜇0,𝑖 −1 = 1 𝜇0,𝑝ℎ + 1 𝜇0,𝑖𝑜𝑛 + 1 𝜇0,𝑔𝑏 + ⋯ 𝜇𝐻 𝜂 = 𝜇0 3 2 + 2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 + 𝑟 𝐹𝑟+ Τ 1 2 𝜂 𝑛 𝜂 = 4𝜋 2𝑚𝑑 ∗ 𝑘𝐵 𝑇 ℎ2 Τ 3 2 𝐹𝑟+1 𝜂
  24. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    24 Electrical conductivity Now it is possible to calculate the electrical conductivity (don’t forget that 𝜇 𝜂 = Τ 𝜇𝐻 𝜂 𝑟𝐻 𝜂 ): 𝜎 𝜂 = 𝑒𝑛 𝜂 𝜇 𝜂 = 8𝜋𝑒 2𝑚𝑑 ∗ 𝑘𝐵 𝑇 Τ 3 2 3ℎ3 𝜇0 𝐹𝑟+ Τ 1 2 𝜂 = 𝜎𝐸0 ln 1 + 𝑒𝜂 here 𝜎𝐸0 is the magnitude of conductivity for a given 𝜂 (describes the conductive “quality” of charge carriers in the material) called transport parameter. 𝜎𝐸0 = 8𝜋𝑒 2𝑚𝑒 𝑘𝐵 𝑇 Τ 3 2 3ℎ3 𝜇0 𝑚𝑑 ∗ 𝑚𝑒 Τ 3 2 = 8𝜋𝑒 2𝑚𝑒 𝑘𝐵 𝑇 Τ 3 2 3ℎ3 𝜇𝑤 𝜇𝑤 = 𝜇0 𝑚𝑑 ∗ 𝑚𝑒 Τ 3 2 is the weighted mobility Reminder: 𝜇0 = Τ 𝑒𝜏0 𝑚𝐼 ∗ and 𝑚𝑑 ∗ ≈ 𝑚𝑆 ∗ for acoustic phonon scattering.
  25. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    25 Electrical conductivity 𝜎 𝜂 = 𝜎𝐸0 ln 1 + 𝑒𝜂 , 𝜎𝐸0 = 8𝜋𝑒 2𝑚𝑒𝑘𝐵𝑇 Τ 3 2 3ℎ3 𝜇0 𝑚𝑑 ∗ 𝑚𝑒 Τ 3 2 , 𝜇𝐻 𝜂 = 𝜇0 3 2 +2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 +𝑟 𝐹𝑟+ Τ 1 2 𝜂 , 𝜇0 = 𝑒𝜋ℏ4𝐶𝑙𝑙 2 𝑘𝐵𝑇 Τ 3 2 𝑚𝑏 ∗ Τ 3 2 𝑚𝐼 ∗Δ𝑑𝑒𝑓 2
  26. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    26 Thermoelectric efficiency Using calculated 𝑚∗, 𝜇0, and 𝜅𝑙 it is possible to calculate a ‘theoretical’ 𝑧𝑇 to estimate the optimum carrier density at a particular temperature (from the plot of the 𝑧𝑇 versus 𝑛): 𝑧𝑇 = 𝛼2𝜎𝑇 𝜅𝑙 + 𝜅𝑒 = 𝛼2 𝜅𝑙 𝜎𝑇 + 𝐿 Considering all that we know: 𝑧𝑇 𝜂 = 𝛼2 𝜂 𝜅𝑙 𝑇𝜎𝐸0 ln 1 + 𝑒𝜂 + 𝐿 𝜂 = 𝛼2 𝜂 𝜓 𝜂 𝛽 −1 + 𝐿 𝜂 here 𝜓 𝜂 = 8𝜋𝑒 3 2𝑚𝑒 𝑘𝐵 ℎ2 Τ 3 2 𝐹𝑟+ Τ 1 2 𝜂 and 𝛽 = 𝜇0 ൗ 𝑚∗ 𝑚𝑒 Τ 3 2 𝑇 Τ 5 2 𝑘𝑙
  27. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    27 Summary
  28. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    28 Quality factor Again: 𝑧𝑇 𝜂 = 𝛼2 𝜂 𝜅𝑙 𝑇𝜎𝐸0 ln 1 + 𝑒𝜂 + 𝐿 𝜂 = 𝛼2 𝜂 Τ 𝑘𝐵 𝑒 2 𝐵ln 1 + 𝑒𝜂 + 𝐿 𝜂 𝐵 = 𝑘𝐵 𝑒 2 𝜎𝐸0 𝜅𝑙 𝑇 = 𝑘𝐵 𝑒 2 8𝜋𝑒 2𝑚𝑒𝑘𝐵 Τ 3 2 3ℎ3 𝜇𝑤 𝜅𝑙 𝑇 Τ 5 2 is the quality factor. For example, for acoustic phonon scattering 𝜏0 ∝ Τ 1 𝑚𝑏 ∗ Τ 3 2 and considering that 𝑚𝑑 ∗ = 𝑁𝑣 Τ 2 3𝑚𝑏 ∗ , 𝜇0 = 𝑒𝜏0 𝑚𝐼 ∗ the quality factor 𝐵 ∝ Τ 𝑁𝑣 𝑚𝐼 ∗𝜅𝑙 For more details see Ref. [6].
  29. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    29 Electronic quality factor 𝐵𝐸 = 𝑘𝐵 𝑒 2 𝜎𝐸0 is the electronic quality factor related to the transport parameter.[7] Reminder: 𝜎𝐸0 = 8𝜋𝑒 2𝑚𝑒𝑘𝐵𝑇 Τ 3 2 3ℎ3 𝜇0 𝑚𝑑 ∗ 𝑚𝑒 Τ 3 2 is a material constant for a good thermoelectric because the mobility tends to depend as 𝑇− Τ 3 2, which cancels the temperature dependence here.
  30. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    30 Summary 𝑧𝑇 𝜂 = 𝛼2 𝜂 Τ 𝑘𝐵 𝑒 2 𝐵ln(1+𝑒𝜂) +𝐿 𝜂 𝛼 𝜂 = ± 𝑘𝐵 𝑒 𝑟+ Τ 5 2 𝐹𝑟+ Τ 3 2 𝜂 𝑟+ Τ 3 2 𝐹𝑟+ Τ 1 2 𝜂 − 𝜂 𝐵 = 𝑘𝐵 𝑒 2 8𝜋𝑒 2𝑚𝑒𝑘𝐵 Τ 3 2 3ℎ3 𝜇𝑤 𝜅𝑙 𝑇 Τ 5 2 𝐿 𝜂 = 𝑘𝐵 𝑒 2 𝑟+ Τ 7 2 𝐹𝑟+ Τ 5 2 𝜂 𝑟+ Τ 3 2 𝐹𝑟+ Τ 1 2 𝜂 − 𝑟+ Τ 5 2 𝐹𝑟+ Τ 3 2 𝜂 𝑟+ Τ 3 2 𝐹𝑟+ Τ 1 2 𝜂 2 At the same time 𝑛 𝜂 = 4𝜋 2𝑚𝑑 ∗ 𝑘𝐵𝑇 ℎ2 Τ 3 2 𝐹𝑟+1 𝜂 𝑛𝐻 𝜂 = 𝑛 𝜂 𝑟𝐻 𝜂 , 𝑟𝐻 𝜂 = 3 2 𝐹 Τ 1 2 𝜂 3 2 +2𝑟 𝐹2𝑟+ Τ 1 2 𝜂 3 2 +𝑟 2 𝐹𝑟+ Τ 1 2 2 𝜂 𝑧𝑇 as function of doping requires only 𝜇𝑤 and 𝜅𝑙. It is enough to measure one sample (𝛼, 𝜎, 𝜅 and 𝑛𝐻) to draw the entire 𝑧𝑇 = 𝑓 𝑛𝐻 curve.
  31. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    31 How to calculate? One way is to use all the formulas that were presented in this and previous lecture as it is (assuming acoustic phonon scattering 𝑟 = − Τ 1 2, for example). Thus, the value of 𝜂 can be found from the experimental value of 𝛼 considering 𝛼 𝜂 = ± 𝑘𝐵 𝑒 𝑟+ Τ 5 2 𝐹𝑟+ Τ 3 2 𝜂 𝑟+ Τ 3 2 𝐹𝑟+ Τ 1 2 𝜂 − 𝜂 . After this all the transport parameters can be calculated. Another possible way to perform the SPB calculations is to use approximated sigmoid functions proposed by Snyder’s group.[7–9]
  32. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    32 Calculations Assuming acoustic phonon scattering (which is reasonable for many of thermoelectrics):[7] 𝛼 𝜂 = ± 𝑘𝐵 𝑒 2𝐹1 𝜂 𝐹0 𝜂 − 𝜂 thus 𝛼𝑟 = 𝛼 Τ 𝑘𝐵 𝑒 = 2𝐹1 𝜂 𝐹0 𝜂 − 𝜂 at the same time: 𝜎 = 𝜎𝐸0 𝐹0 𝜂 = 𝐵𝐸 𝑒 𝑘𝐵 2 𝐹0 𝜂 while 𝐵𝐸 = 𝑘𝐵 𝑒 2 𝜎𝐸0 = 𝑘𝐵 𝑒 2 𝛼2𝜎𝜎𝐸0 𝛼2𝜎 = 𝑘𝐵 𝑒 2 𝛼2𝜎𝜎𝐸0 𝛼2𝜎𝐸0 𝐹0 𝜂 = 𝛼2𝜎 𝛼2 Τ 𝑘𝐵 𝑒 2 𝐹0 𝜂 = 𝛼2𝜎 2𝐹1 𝜂 𝐹0 𝜂 −𝜂 2 𝐹0 𝜂 = 𝛼2𝜎 𝛼𝑟 2𝐹0 𝜂 For non-degenerate limit 𝐹0 𝜂 reduces to 𝐹0 𝜂 = 𝑒2−𝛼𝑟 In fully degenerate case 𝐹0 𝜂 reduces to 𝐹0 𝜂 = 𝜋2 3𝛼𝑟
  33. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    33 Calculations Thus, by using sigmoid selection function that smoothly goes between the degenerate and non- degenerate limits the following relation can be written for 𝐵𝐸:[7,8] 𝐵𝐸 = 𝛼2𝜎 𝛼𝑟 2𝑒2−𝛼𝑟 1 + 𝑒−5 𝛼𝑟−1 + 𝛼𝑟 Τ 𝜋2 3 1 + 𝑒5 𝛼𝑟−1 −1 here 𝛼𝑟 = 𝛼 Τ 𝑘𝐵 𝑒 , where 𝛼 is the experimental value of the Seebeck coefficient.
  34. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    34 Calculations Similarly, the electrical conductivity:[7,8] 𝜎 = 𝜎𝐸0 𝑒2−𝛼𝑟 1 + 𝑒−5 𝛼𝑟−1 + Τ 𝜋2 3𝛼𝑟 1 + 𝑒5 𝛼𝑟−1 here 𝛼𝑟 = 𝛼 Τ 𝑘𝐵 𝑒 , where 𝛼 is the experimental value of the Seebeck coefficient. In this case the reduced Fermi energy can be calculated from 𝜎 = 𝜎𝐸0 ln 1 + 𝑒𝜂
  35. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    35 Calculations In the same manner (see details in Refs. [7–9]) such relation can be written for the Seebeck effective mass: 𝑚𝑆 ∗ = ℎ2 2𝑘𝐵 𝑇 3𝑛𝐻 16 𝜋 Τ 2 3 𝑒𝛼𝑟−2 − 0.17 Τ 2 3 1 + 𝑒−5 𝛼𝑟−1 + 3 𝜋2 2 𝜋 Τ 2 3 𝛼𝑟 1 + 𝑒5 𝛼𝑟−1 and for the weighted mobility: 𝜇𝑤 = 3ℎ3𝜎 8𝜋𝑒 2𝑚𝑒 𝑘𝐵 𝑇 Τ 3 2 𝑒𝛼𝑟−2 1 + 𝑒−5 𝛼𝑟−1 + 3 𝜋2 𝛼𝑟 1 + 𝑒5 𝛼𝑟−1 here 𝛼𝑟 = 𝛼 Τ 𝑘𝐵 𝑒 , where 𝛼 is the experimental value of the Seebeck coefficient.
  36. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    36 Hall and weighted mobilities 𝜇𝐻 = 𝜎𝑅𝐻 = 𝜇0 𝐹− Τ 1 2 2𝐹0 𝜇𝑤 = 𝑓 𝛼, 𝜎 = 𝜇0 𝑚𝑑 ∗ 𝑚𝑒 3 2 𝜇𝐻 and 𝜇𝑤 are highly correlated except in samples with very high carrier concentration. This is because the Hall mobility depends on Fermi level through the Fermi integrals. This reduces the 𝜇𝐻 at high carrier concentrations. The weighted mobility should only depend on the intrinsic mobility parameter and the density of states 𝑚∗, which shouldn’t change much in a parabolic band.[8]
  37. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    37 Hall and weighted mobilities Often the Hall mobility measurements are not accurate enough to notice the trend. The weighted mobility, in contrast, only depends on the mobility parameter and not Fermi level, so it is reasonable to expect it to be a constant with carrier concentration.[8] 𝜇𝐻 = 𝜇0 𝐹− Τ 1 2 2𝐹0 𝜇𝑤 = 𝜇0 𝑚𝑑 ∗ 𝑚𝑒 3 2 𝑚𝑑 ∗ 𝑚𝑒 = 𝜇𝑤 𝜇0 2 3 = 𝜇𝑤 𝜇𝐻 𝐹− Τ 1 2 2𝐹0 2 3
  38. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    38 References 1. Kasap, S.; Koughia, C.; Ruda, H.E. Electrical Conduction in Metals and Semiconductors. In Springer Handbook of Electronic and Photonic Materials; Springer International Publishing: Cham, 2017; pp 19–45. 2. Kireev, P.S. Semiconductor Physics, 2nd ed.; Mir: Moscow, 1978. 3. Matthiessen, A.; von Bose, M. On the Influence of Temperature on the Electric Conducting Power of Metals. Philos. Trans. R. Soc. London 1862, 152, 1–27. 4. Grundmann, M. The Physics of Semiconductors; Graduate Texts in Physics; Springer International Publishing: Cham, 2016. 5. Fu, C.; Zhu, T.; Pei, Y.; Xie, H.; Wang, H.; Snyder, G.J.; Liu, Y.; Liu, Y.; Zhao, X. High Band Degeneracy Contributes to High Thermoelectric Performance in P-Type Half-Heusler Compounds. Adv. Energy Mater. 2014, 4 (18), 1400600. 6. Zevalkink, A.; Smiadak, D.M.; Blackburn, J.L.; Ferguson, A.J.; Chabinyc, M.L.; Delaire, O.; Wang, J.; Kovnir, K.; Martin, J.; Schelhas, L.T.; Sparks, T.D.; Kang, S.D.; Dylla, M.T.; Snyder, G.J.; Ortiz, B.R.; Toberer, E.S. A Practical Field Guide to Thermoelectrics: Fundamentals, Synthesis, and Characterization. Appl. Phys. Rev. 2018, 5 (2), 021303.
  39. Лекция «Введение в физику полупроводников» / 6 августа 2022 г.

    39 References 7. Zhang, X.; Bu, Z.; Shi, X.; Chen, Z.; Lin, S.; Shan, B.; Wood, M.; Snyder, A.H.; Chen, L.; Snyder, G.J.; Pei, Y. Electronic Quality Factor for Thermoelectrics. Sci. Adv. 2020, 6 (46), eabc0726. 8. Snyder, G.J.; Snyder, A.H.; Wood, M.; Gurunathan, R.; Snyder, B.H.; Niu, C. Weighted Mobility. Adv. Mater. 2020, 32 (25), 2001537. 9. Snyder, G.J.; Pereyra, A.; Gurunathan, R. Effective Mass from Seebeck Coefficient. Adv. Funct. Mater. 2022, 32 (20), 2112772. This work was inspired by brilliant course on Principles of Thermoelectric Materials Engineering by prof. Jeffrey G. Snyder (Northwestern University, USA) in the framework of On-Demand Seminar “Introduction to Thermoelectric Conversion” (February 2021).