Skip to main content
SpringerLink
Log in

A Theoretical Model of Thermoelectric Transport Properties for Electrons and Phonons

  • Published:
Journal of Electronic Materials Aims and scope Submit manuscript

Abstract

A generic theoretical model for five bulk thermoelectric materials (PbTe, Bi2Te3, SnSe, Si0.7Ge0.3, and Mg2Si) has been developed based on the semiclassical model incorporating nonparabolicity, the two-band Kane model, the Hall factor, and the Debye–Callaway model for electrons and phonons. It is used to calculate thermoelectric transport properties, viz. the Seebeck coefficient, electrical conductivity, and electronic and lattice thermal conductivities, in the temperature range from room temperature up to 1200 K. The present model differs from others in the following regards: Firstly, thorough verification of modified electron scattering mechanisms is carried out by comparison with reported experimental data; Secondly, extensive verification of the model is presented, with concomitant agreement between calculations and reported measurements of effective masses, electron and hole concentrations, Seebeck coefficient, electrical conductivity, and electronic and lattice thermal conductivities; Thirdly, the present model provides the Fermi energy as a function of temperature and doping concentration; Fourthly, the velocities of sound are calculated using the Debye model rather than taken from literature. After verification of the present model, we were able to examine the recently attractive material SnSe, indicating a significant improvement in the dimensionless figure of merit.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.M. Rowe, CRC Handbook of Thermoelectrics (Boca Raton: CRC Press, 1995), p. 539.

    Book  Google Scholar 

  2. D.M. Rowe, Thermoelectrics Handbook; Macro to Nano, Vol. 56 (Boca Raton: CRC Taylor & Francis, 2006).

    Google Scholar 

  3. G. Shi and E. Kioupakis, J. Appl. Phys. 117, 065103 (2015).

    Article  Google Scholar 

  4. L. Hicks and M. Dresselhaus, Phys. Rev. B 47, 12727 (1993).

    Article  Google Scholar 

  5. T.C. Harman, D.L. Spears, and M.J. Manfra, J. Electron. Mater. 25, 1121 (1996).

    Article  Google Scholar 

  6. J. Zhou, X. Li, G. Chen, and R. Yang, Phys. Rev. B 82, 115308 (2010).

    Article  Google Scholar 

  7. P. Debye and E. Conwell, Phys. Rev. 93, 693 (1954).

    Article  Google Scholar 

  8. C. Herring and E. Vogt, Phys. Rev. 101, 944 (1956).

    Article  Google Scholar 

  9. R.P. Chasmar and R. Stratton, J. Electron. Control 7, 52 (1959).

    Article  Google Scholar 

  10. J.P. Dismukes, L. Ekstrom, E.F. Steigmeier, I. Kudman, and D.S. Beers, J. Appl. Phys. 35, 2899 (1964).

    Article  Google Scholar 

  11. F.D. Rosi, Solid State Electron. 11, 833 (1968).

    Article  Google Scholar 

  12. Y.I. Ravich, B.A. Efimova, and I.A. Smirnov, Semiconducting Lead Chalcogenides (New York: Plenum Press, 1970).

    Book  Google Scholar 

  13. Y.I. Ravich, B.A. Efimova, and V.I. Tamabohenko, Phys. Stat. Sol. 43, 453 (1971).

    Article  Google Scholar 

  14. J.J. Harris and B.K. Ridley, J. Phys. Chem. Solids 33, 1455 (1972).

    Article  Google Scholar 

  15. C.B. Vining, J. Appl. Phys. 69, 331 (1991).

    Article  Google Scholar 

  16. M. Lundstrom, Fundamentals of Carrier Transport, 2nd ed. (Cambridge: Cambridge University Press, 2000).

    Book  Google Scholar 

  17. A. Minnich and G. Chen, Appl. Phys. Lett. 91, 073105 (2007).

    Article  Google Scholar 

  18. A. Minnich, H. Lee, X. Wang, G. Joshi, M. Dresselhaus, Z. Ren, G. Chen, and D. Vashaee, Phys. Rev. B 80, 155327 (2009).

    Article  Google Scholar 

  19. G. Chen, Nanoscale Energy Transport and Conversion (Oxford: Oxford University Press, 2005), p. 236.

    Google Scholar 

  20. C. Vineis, T. Harman, S. Calawa, M. Walsh, R. Reeder, R. Singh, and A. Shakouri, Phys. Rev. B 77, 235202 (2008).

    Article  Google Scholar 

  21. S. Youn and A. Freeman, Phys. Rev. B 63, 085112 (2001).

    Article  Google Scholar 

  22. M. Kim, A. Freeman, and C. Geller, Phys. Rev. B 72, 035205 (2005).

    Article  Google Scholar 

  23. D. Bilc, S. Mahanti, and M. Kanatzidis, Phys. Rev. B 74, 125202 (2006).

    Article  Google Scholar 

  24. J.-H. Bahk, Z. Bian, and A. Shakouri, Phys. Rev. B 89, 075204 (2014).

    Article  Google Scholar 

  25. H.-W. Jeon, H.-P. Ha, D.-B. Hyun, and J.-D. Shim, J. Phys. Chem. Solids 52, 579 (1991).

    Article  Google Scholar 

  26. Y. Pei, A.D. LaLonde, H. Wang, and G.J. Snyder, Energy Environ. Sci. 5, 7963 (2012).

    Article  Google Scholar 

  27. J.-I. Tani and H. Kido, Phys. B 364, 218 (2005).

    Article  Google Scholar 

  28. L.D. Zhao, S.H. Lo, Y. Zhang, H. Sun, G. Tan, C. Uher, C. Wolverton, V.P. Dravid, and M.G. Kanatzidis, Nature 508, 373 (2014).

    Article  Google Scholar 

  29. A.H. Wilson, The Theory of Metals, 2nd ed. (Cambridge: Cambridge University Press, 1953), p. 4.

    Google Scholar 

  30. J.M. Ziman, Electrons and Phonons (London: Oxford University Press, 1960), p. 265.

    Google Scholar 

  31. J. Callaway, Phys. Rev. 113, 1046 (1959).

    Article  Google Scholar 

  32. M. Holland, Phys. Rev. 132, 2461 (1963).

    Article  Google Scholar 

  33. C. Herring, Bell Syst. Tech. J. 34, 237 (1955).

    Article  Google Scholar 

  34. E.O. Kane, Semiconductors and Semimetals, Chapter 3 The k–p Method (New York: Academic Press, 1966).

    Google Scholar 

  35. E.H. Putley, The Hall Effect and Related Phenomena (London: Butterworths, 1960).

    Google Scholar 

  36. B. Abeles and S. Meiboom, Phys. Rev. 95, 31 (1954).

    Article  Google Scholar 

  37. M. Shibuya, Phys. Rev. 95, 1385 (1954).

    Article  Google Scholar 

  38. J. Heremans, C. Thrush, and D. Morelli, Phys. Rev. B 70, 115334 (2004).

    Article  Google Scholar 

  39. H.J. Goldsmid, Thermoelectric Refrigeration (New York: Plenum Press, 1964), p. 39.

    Book  Google Scholar 

  40. G.S. Nolas, J. Sharp, and H.J. Goldsmid, Thermoelectrics: Basic Principles and New Materials Developments (Berlin: Springer, 2001).

    Book  Google Scholar 

  41. N.F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys (New York: Dover Publications, 1958), p. 310.

    Google Scholar 

  42. J.P. Heremans, B. Wiendlocha, and A.M. Chamoire, Energy Environ. Sci. 5, 5510 (2012).

    Article  Google Scholar 

  43. J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950).

    Article  Google Scholar 

  44. H. Brooks, Phys. Rev. 83, 879 (1951).

    Google Scholar 

  45. D.J. Howarth and E.H. Sondheimer, Proc. R. Soc. Lond. 219, 53 (1953).

    Article  Google Scholar 

  46. H. Ehrenreich, J. Phys. Chem. Solids 2, 131 (1957).

    Article  Google Scholar 

  47. H. Ehrenreich, J. Appl. Phys. 32, 2155 (1961).

    Article  Google Scholar 

  48. Y.I. Ravich, B.A. Efimova, and V.I. Tamabohenko, Phys. Stat. Sol. 43, 11 (1971).

    Article  Google Scholar 

  49. B.R. Nag, Electron Transport in Compound Semiconductors (New York: Springer, 1980), p. 118.

    Book  Google Scholar 

  50. D.M. Zayachuk, Semiconductors 31, 173 (1997).

    Article  Google Scholar 

  51. D.M. Freik, L.I. Nykyruy, and V.M. Shperun, Semicond. Phys. Quantum Electron. Optoelectron. 5, 362 (2002).

    Google Scholar 

  52. B.-L. Huang and M. Kaviany, Phys. Rev. B 77, 125209 (2008).

    Article  Google Scholar 

  53. S. Ahmad and S.D. Mahanti, Phys. Rev. B 81, 165203 (2010).

    Article  Google Scholar 

  54. D.A. Broido and T.L. Reinecke, Appl. Phys. Lett. 70, 2834 (1997).

    Article  Google Scholar 

  55. J. Kolodziejczak, Phys. Stat. Sol. 19, 231 (1967).

    Article  Google Scholar 

  56. A. Amith, I. Kudman, and E. Steigmeier, Phys. Rev. 138, A1270 (1965).

    Article  Google Scholar 

  57. S.K. Bux, M.T. Yeung, E.S. Toberer, G.J. Snyder, R.B. Kaner, and J.-P. Fleurial, J. Mater. Chem. 21, 12259 (2011).

    Article  Google Scholar 

  58. C.-L. Chen, H. Wang, Y.-Y. Chen, T. Day, and G.J. Snyder, J. Mater. Chem. A 2, 11171 (2014).

    Article  Google Scholar 

  59. L.M. Rogers, Br. J. Appl. Phys. 18, 1227 (1967).

    Article  Google Scholar 

  60. J. Androulakis, Y. Lee, I. Todorov, D.-Y. Chung, and M. Kanatzidis, Phys. Rev. B 83, 195203 (2011).

    Article  Google Scholar 

  61. H. Chi, W. Liu, K. Sun, X. Su, G. Wang, P. Lošt’ák, V. Kucek, Č. Drašar, and C. Uher, Phys. Rev. B 88, 045202 (2013).

    Article  Google Scholar 

  62. P. Debye, Ann. Phys. Leipz. 39, 798 (1912).

    Google Scholar 

  63. H. Callen, Phys. Rev. 76, 1394 (1949).

    Article  Google Scholar 

  64. E. Conwell and V. Weisskopf, Phys. Rev. 77, 388 (1950).

    Article  Google Scholar 

  65. F.J. Blatt, Solid State Physics, Vol. 4, ed. F. Seitz and D. Turnbull (New York: Academic Press, 1957), p. 344.

    Google Scholar 

  66. D. Chattopadhyay and H. Queisser, Rev. Mod. Phys. 53, 745 (1981).

    Article  Google Scholar 

  67. C. Herring, Phys. Rev. 95, 954 (1954).

    Article  Google Scholar 

  68. P.G. Klemens, Proc. Phys. Soc. 68, 1113 (1955).

    Article  Google Scholar 

  69. J.M. Ziman, Philos. Mag. 1, 191 (1956).

    Article  Google Scholar 

  70. E. Steigmeier and B. Abeles, Phys. Rev. 136, A1149 (1964).

    Article  Google Scholar 

  71. R. Berman, Thermal Conduction in Solids (Oxford: Clarendon Press, 1976), p. 30.

    Google Scholar 

  72. G. Leibfried and E. Schlomann, Nachrichten der Akademie der Wissenschaften in Gottingen, lia, Mathematisch-Physikalische Klasse, 71 (1954).

  73. J.M. Ziman, Philos. Mag. 2, 292 (1956).

    Article  Google Scholar 

  74. H.J. Goldsmid, Proc. Phys. Soc. 72, 17 (1958).

    Article  Google Scholar 

  75. P. Klemens, Phys. Rev. 119, 507 (1960).

    Article  Google Scholar 

  76. G. Slack and C. Glassbrenner, Phys. Rev. 120, 782 (1960).

    Article  Google Scholar 

  77. B. Abeles, D. Beers, G. Cody, and J. Dismukes, Phys. Rev. 125, 44 (1962).

    Article  Google Scholar 

  78. M. Holland, Phys. Rev. 134, A471 (1964).

    Article  Google Scholar 

  79. G.T. Alekseeva, B.A. Efimova, L.M. AOstrovskaya, O.S. Serebryannikova, and M.I. Tsypin, Sov. Phys. Semicond. 4, 1122 (1971).

    Google Scholar 

  80. J.E. Parrott, Rev. Int. Ht. Temp. Refract. Fr. 16, 393 (1979).

    Google Scholar 

  81. P.G. Klemens, Thermal Conductivity and Lattice Vibrational Modes, Chapter 1, Solid State Physics, 7th ed. (New York: Academic Press, 1958), p. 41.

    Google Scholar 

  82. N.W. Ashcroft and N.D. Mermin, Solid State Physics (New York: Holt, Rinehart, and Winston, 1976), p. 323.

    Google Scholar 

  83. A.F. Ioffe, Semiconductor Thermoelement and Thermoelectric Cooling (London: Infosearch Limited, 1957), p. 52.

    Google Scholar 

  84. B. Huang, C. Lawrence, A. Gross, G.-S. Hwang, N. Ghafouri, S.-W. Lee, H. Kim, C.-P. Li, C. Uher, K. Najafi, and M. Kaviany, J. Appl. Phys. 104, 113710 (2008).

    Article  Google Scholar 

  85. N. Satyala and D. Vashaee, J. Electron. Mater. 41, 1785 (2012).

    Article  Google Scholar 

  86. J.-I. Tani and H. Kido, J. Alloys Compd. 466, 335 (2008).

    Article  Google Scholar 

  87. C. Kittel, Introduction to Solid State Physics, 8th ed. (New York: Wiley, 2005), p. 21.

    Google Scholar 

  88. O. Madelung, Numerical Data and Functional Relationships in Science and Technology, Group III: Crystal and Solid Physics (Springer, Berlin, 1983), p. 276.

    Google Scholar 

  89. B. Abeles, Phys. Rev. 131, 1906 (1963).

    Article  Google Scholar 

  90. H. Lyden, Phys. Rev. 135, A514 (1964).

    Article  Google Scholar 

  91. H. Kohler, Phys. Stat. Sol. 73, 95 (1976).

    Article  Google Scholar 

  92. X. He, H. Shen, W. Wang, Z. Wang, B. Zhang, and X. Li, J. Alloys Compd. 556, 86 (2013).

    Article  Google Scholar 

  93. G.A. Slack and M.A. Hussain, J. Appl. Phys. 70, 2694 (1991).

    Article  Google Scholar 

  94. M. Fischetti and S. Laux, Phys. Rev. B 48, 2244 (1993).

    Article  Google Scholar 

  95. O.A. Pankratov and P.P. Povarov, Solid State Commun. 66, 847 (1988).

    Article  Google Scholar 

  96. D.H. Parkinson and J.E. Quarrington, Proc. Phys. Soc. 67, 569 (1954).

    Article  Google Scholar 

  97. R. Allgaier and W. Scanlon, Phys. Rev. 111, 1029 (1958).

    Article  Google Scholar 

  98. R.S. Allgaier, J. Appl. Phys. 32, 2185 (1961).

    Article  Google Scholar 

  99. W. Cochran, R.A. Cowley, G. Dolling, and M.M. Elcombe, Proc. R. Soc. Lond. 293, 433 (1966).

    Article  Google Scholar 

  100. D.A. Wright, Metall. Rev. 15, 147 (1970).

    Google Scholar 

  101. Y. Tsang and M. Cohen, Phys. Rev. B 3, 1254 (1971).

    Article  Google Scholar 

  102. A.J. Crocker and L.M. Rogers, Br. J. Appl. Phys. 18, 563 (1967).

    Article  Google Scholar 

  103. G. Martinez, M. Schlüter, and M. Cohen, Phys. Rev. B 11, 660 (1975).

    Article  Google Scholar 

  104. P. Boulet, M.J. Verstraete, J.P. Crocombette, M. Briki, and M.C. Record, Comput. Mater. Sci. 50, 847 (2011).

    Article  Google Scholar 

  105. Y. Zhang, X. Ke, C. Chen, J. Yang, and P. Kent, Phys. Rev. B. 80, 024304 (2009).

    Article  Google Scholar 

  106. Z. Tian, K. Esfarjani, J. Shiomi, A.S. Henry, and G. Chen, Appl. Phys. Lett. 99, 053122 (2011).

    Article  Google Scholar 

  107. Z. Tian, J. Garg, K. Esfarjani, T. Shiga, J. Shiomi, and G. Chen, Phys. Rev. B 85, 184303 (2012).

    Article  Google Scholar 

  108. B.-T. Wang and P. Zhang, Appl. Phys. Lett. 100, 082109 (2012).

    Article  Google Scholar 

  109. D. Bilc, S. Mahanti, E. Quarez, K.-F. Hsu, R. Pcionek, and M. Kanatzidis, Phys. Rev. Lett. 93, 146403 (2004).

    Article  Google Scholar 

  110. G. Martinez, M. Schlüter, and M. Cohen, Phys. Rev. B. 11, 651 (1975).

    Article  Google Scholar 

  111. T. Shiga, J. Shiomi, J. Ma, O. Delaire, T. Radzynski, A. Lusakowski, K. Esfarjani, and G. Chen, Phys. Rev. B. 85, 155203 (2012).

    Article  Google Scholar 

  112. E.F. Steigmeier, Appl. Phys. Lett. 3, 6 (1963).

    Article  Google Scholar 

  113. H.J. Goldsmid and R.W. Douglas, Br. J. Appl. Phys. 5, 386 (1954).

    Article  Google Scholar 

  114. T.C. Harman, B. Paris, S.E. Miller, and H.L. Goering, J. Phys. Chem. Solids 2, 181 (1957).

    Article  Google Scholar 

  115. J.R. Drabble and C.H.L. Goodman, J. Phys. Chem. Solids 5, 142 (1958).

    Article  Google Scholar 

  116. H. Kohler, Phys. Stat. Sol. 74, 591 (1976).

    Article  Google Scholar 

  117. M.S. Park, J.-H. Song, J.E. Medvedeva, M. Kim, I.G. Kim, and A.J. Freeman, Phys. Rev. B 81, 155211 (2010).

    Article  Google Scholar 

  118. J. Jenkins, J. Rayne, and R. Ure, Phys. Rev. B 5, 3171 (1972).

    Article  Google Scholar 

  119. S.K. Mishra, S. Satpathy, and O. Jepsen, J. Phys. 9, 461 (1997).

    Google Scholar 

  120. P. Larson, S.D. Mahanti, and M.G. Kanatzidis, Phys. Rev. B 61, 8162 (2000).

    Article  Google Scholar 

  121. J. Merkisz, P. Fuc, P. Lijewski, A. Ziolkowski, and K.T. Wojciechowski, J. Electron. Mater. 44, 1704 (2014).

    Article  Google Scholar 

  122. H. Rauh, R. Geick, H. Kohler, N. Nucker, and N. Lehner, J. Phys. C 14, 2705 (1981).

    Article  Google Scholar 

  123. D. Bessas, I. Sergueev, H.C. Wille, J. Perßon, D. Ebling, and R.P. Hermann, Phys. Rev. B 86, 224301 (2012).

    Article  Google Scholar 

  124. J.D. Wasscher, W. Albers, and C. Haas, Solid State Electron. 6, 261 (1963).

    Article  Google Scholar 

  125. K. Adouby, C. Perez-Vicente, and J.C. Jumas, Z. Kristallogr. 213, 343 (1998).

    Google Scholar 

  126. B.B. Nariya, A.K. Dasadia, M.K. Bhayani, A.J. Patel, and A.R. Jani, Chalcogenide Lett. 6, 549 (2009).

    Google Scholar 

  127. A. Banik and K. Biswas, J. Mater. Chem. A 2, 9620 (2014).

    Article  Google Scholar 

  128. S. Sassi, C. Candolfi, J.B. Vaney, V. Ohorodniichuk, P. Masschelein, A. Dauscher, and B. Lenoir, Appl. Phys. Lett. 104, 212105 (2014).

    Article  Google Scholar 

  129. Y.-M. Han, J. Zhao, M. Zhou, X.-X. Jiang, H.-Q. Leng, and L.-F. Li, J. Mater. Chem. A 3, 4555 (2015).

    Article  Google Scholar 

  130. M. Au-Yang and M. Cohen, Phys. Rev. 178, 1279 (1969).

    Article  Google Scholar 

  131. R. Car, G. Ciucci, and L. Quartapelle, Phys. Stat. Sol. (b) 86, 471 (1978).

    Article  Google Scholar 

  132. L. Makinistian and E.A. Albanesi, Phys. Status Solidi 246, 183 (2009).

    Article  Google Scholar 

  133. S. Chen, K. Cai, and W. Zhao, Phys. B 407, 4154 (2012).

    Article  Google Scholar 

  134. Y. Sun, Z. Zhong, T. Shirakawa, C. Franchini, D. Li, Y. Li, S. Yunoki, and X.-Q. Chen, Phys. Rev. B 88, 235122 (2013).

    Article  Google Scholar 

  135. J. Carrete, N. Mingo, and S. Curtarolo, Appl. Phys. Lett. 105, 101907 (2014).

    Article  Google Scholar 

  136. J.E. Parrott, Proc. Phys. Soc. 81, 726 (1963).

    Article  Google Scholar 

  137. A. Amith, Phys. Rev. 139, A1624 (1965).

    Article  Google Scholar 

  138. N. Gaur, C. Bhandari, and G. Verma, Phys. Rev. 144, 628 (1966).

    Article  Google Scholar 

  139. P. Koenig, D.W. Lynch, and G.C. Danielson, J. Phys. Chem. Solids 20, 122 (1961).

    Article  Google Scholar 

  140. P. Lee, Phys. Rev. 135, A1110 (1964).

    Article  Google Scholar 

  141. M. Au-Yang and M. Cohen, Phys. Rev. 178, 1358 (1969).

    Article  Google Scholar 

  142. A.V. Krivosheeva, A.N. Kholod, V.L. Shaposhnikov, A.E. Krivosheev, and V.E. Borisenko, Semiconductors 36, 496 (2002).

    Article  Google Scholar 

  143. J.J. Martin, J. Phys. Chem. Solids 33, 1139 (1972).

    Article  Google Scholar 

  144. M. Akasaka, T. Iida, K. Nishio, and Y. Takanashi, Thin Solid Films 515, 8237 (2007).

    Article  Google Scholar 

  145. H. Wang, H. Jin, W. Chu, and Y. Guo, J. Alloys Compd. 499, 68 (2010).

    Article  Google Scholar 

  146. K. Kutorasiński, J. Tobola, and S. Kaprzyk, Phys. Rev. B 87, 195205 (2013).

    Article  Google Scholar 

  147. X. Zhang, H. Liu, Q. Lu, J. Zhang, and F. Zhang, Appl. Phys. Lett. 103, 063901 (2013).

    Article  Google Scholar 

  148. N. Farahi, M. VanZant, J. Zhao, J.S. Tse, S. Prabhudev, G.A. Botton, J.R. Salvador, F. Borondics, Z. Liu, and H. Kleinke, Dalton Trans. 43, 14983 (2014).

    Article  Google Scholar 

  149. V. Zaitsev, M. Fedorov, E. Gurieva, I. Eremin, P. Konstantinov, A. Samunin, and M. Vedernikov, Phys. Rev. B 74, 045207 (2006).

    Article  Google Scholar 

  150. R. Morris, R. Redin, and G. Danielson, Phys. Rev. 109, 1909 (1958).

    Article  Google Scholar 

  151. B.C. Gerstein, J. Chem. Phys. 47, 2109 (1967).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Western Michigan University, Kalamazoo, MI, 49008, USA

    HoSung Lee

Authors
  1. HoSung Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to HoSung Lee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, H. A Theoretical Model of Thermoelectric Transport Properties for Electrons and Phonons. J. Electron. Mater. 45, 1115–1141 (2016). https://doi.org/10.1007/s11664-015-4245-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11664-015-4245-z

Keywords

Search

Navigation