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Wiedemann-Franz Law for Magnon Transport

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Phys. Rev. B 92, 134425 (2015)
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.134425
arXiv:1507.03807
http://arxiv.org/abs/1507.03807

Published in: Science
  • Updated: http://www.slideshare.net/koukiNakata/magnonic-wiedemannfranz-law
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  • Dec. 2016: Updated file is available here; https://www.dropbox.com/s/5n40xudfu51ibj3/MagnonicWFlaw_KoukiNakata.pdf?dl=0
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  • Textbook by Mahan: Chap. 3 (p.180-)
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  • See also http://www.slideshare.net/koukiNakata/suppl-mat-for-magnon-wf-law
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  • Textbook by Ashcroft & Mermin: Eq. (13.56)
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Wiedemann-Franz Law for Magnon Transport

  1. 1. Wiedemann-Franz Law for Magnon Transport Based on [Phys. Rev. B 92, 134425 (2015)] by KN, P. Simon, and D. Loss Kouki Nakata Univ. of Basel All the responsibility of this slide rests with “Kouki Nakata”
  2. 2.  MAIN MESSAGE
  3. 3. 162 YEARS AGO due to electron (Fermion) [R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]  「Wiedemann-Franz Law」 𝜋2 3 𝑘B 𝑒 2 𝑇 Thermoelectric Effects in Metal
  4. 4. THEN
  5. 5. Thermomagnetic Effects in FI QUESTION  How expressed in `AN EQUATION’ ? due to magnon (Boson) Universality
  6. 6. 𝑘B 𝑔𝜇B 2 𝑇 ANSWER [KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
  7. 7. WHY? We discuss from now on
  8. 8.  BACKGROUND
  9. 9. Universal Thermomagnetic Relation of Magnon Transport GOAL  FI:Long-ranged magnetic order  ``Magnon (spin-wave)’’ 𝑘B 𝜇B Magnet Heat ?
  10. 10. Universal Thermomagnetic Relation of Magnon Transport  Thermoelectric properties of Electron transport in metal  Wiedemann-Franz Law Guiding principle  FI:Long-ranged magnetic order  ``Magnon (spin-wave)’’ GOAL
  11. 11. Wiedemann-Franz Law[R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]  Thermoelectric properties of electron transport Lorenz number ℒ ≡ 𝜋2 3 𝑘 𝐵 𝑒 2 : Universal 𝐾 𝜎 = 𝜋2 3 𝑘 𝐵 𝑒 2 𝑇 (𝐾: Thermal conductivity, 𝜎: Electrical conductivity) Low temp.
  12. 12. 𝑗𝑒 𝑗 𝑄 = 𝐿11 𝐿12 𝐿21 𝐿22 𝐸 𝛻𝑇 charge Heat Onsager matrix 𝐿𝑖𝑗 Thermoelectric Effects Electron (metal) Magnon (FI) WF law (Low temp.) 𝐿22 + 𝑂(𝜀 𝐹 −2 ) 𝐿11 ≈ 𝐾 𝜎 = 𝜋2 3 𝑘 𝐵 𝑒 2 𝑇 ? Lorenz number ℒ ≡ 𝜋2 3 𝑘 𝐵 𝑒 2 ? Seebeck 𝑆 & Peltier Π 𝑆 ≡ 𝐿12 /𝐿11 , 𝛱 ≡ 𝐿21 /𝐿11 Thomson relation: 𝛱 = 𝑇𝑆 ?
  13. 13. Electron (metal) Magnon (FI) WF law (Low temp.) 𝐿22 + 𝑂(𝜀 𝐹 −2 ) 𝐿11 ≈ 𝐾 𝜎 = 𝜋2 3 𝑘 𝐵 𝑒 2 𝑇 ? Lorenz number ℒ ≡ 𝜋2 3 𝑘 𝐵 𝑒 2 ? Seebeck 𝑆 & Peltier Π 𝑆 ≡ 𝐿12 /𝐿11 , 𝛱 ≡ 𝐿21 /𝐿11 Thomson relation: 𝛱 = 𝑇𝑆 ? 𝐼m 𝐼 𝑄 = 𝐿11 𝐿12 𝐿21 𝐿22 𝛻𝐵 𝛻𝑇 Magnet Heat Onsager matrix 𝐿𝑖𝑗 Thermomagnetic Effects
  14. 14. 𝐼m 𝐼 𝑄 = 𝐿11 𝐿12 𝐿21 𝐿22 𝛻𝐵 𝛻𝑇 WF Magnet Heat Thermomagnetic Effects Onsager matrix 𝐿𝑖𝑗 Electron (metal) Magnon (FI) WF law (Low temp.) 𝐿22 + 𝑂(𝜀 𝐹 −2 ) 𝐿11 ≈ 𝐾 𝜎 = 𝜋2 3 𝑘 𝐵 𝑒 2 𝑇 𝐾 𝐺 ≡ 𝐿22 − 𝐿21 𝐿12/𝐿11 𝐿11 = ? Lorenz number ℒ ≡ 𝜋2 3 𝑘 𝐵 𝑒 2 ℒm = ? Seebeck 𝑆 & Peltier Π 𝑆 ≡ 𝐿12 /𝐿11 , 𝛱 ≡ 𝐿21 /𝐿11 Thomson relation: 𝛱 = 𝑇𝑆 What is their behaviors at low temp. ?
  15. 15. Charge 𝑒 Magnet 𝜇B Heat 𝑘B TARGET Fermion VS Boson ``Wiedemann-Franz Law’’ [R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)] [KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
  16. 16. Point  Thermal properties “𝒌 𝐁”:Different ? OR Universal ? Magnon Wiedemann-Franz Law  Quantum-statistical properties are different Electron 𝒆 = Fermion Magnon 𝜇B = Boson
  17. 17.  SYSTEM [KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
  18. 18. Ferromagnetic Insulating Junction  𝐽ex ≪ 𝐽 (weak coupling) 𝑇L 𝑇R ∆𝐵 ≡ 𝐵R − 𝐵L ∆𝑇 ≡ 𝑇R − 𝑇L Magnon currents Q. What happen when magnons are in condensation ?  See [PRB 90, 144419 (2014)] & [PRB 92, 014422 (2015)]
  19. 19. Onsager matrix 𝐿𝑖𝑗 Magnetic current Heat current 𝐽ex ≪ 𝐽, ( 𝑎: Lattice constant) ∆𝐵 ≡ 𝐵R − 𝐵L, ∆𝑇 ≡ 𝑇R − 𝑇L 𝑇R 𝑇L Ferromagnetic Insulating Junction 𝐿11 ∝ 𝜇B 2 𝐿22 ∝ 𝑘B 2 𝐿12 ∝ 𝜇B 𝑘B 𝐿21 ∝ 𝜇B 𝑘B
  20. 20.  RESULTS [KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
  21. 21.  Magnon Lorenz number: ℒm ≡ 𝑘 𝐵 𝑔𝜇 𝐵 2 : `Universal’ 𝐾 𝐺 = 𝑘 𝐵 𝑔𝜇 𝐵 2 𝑇 ∝ 𝑇  Thermal magnon conductance: 𝐾 ≡ 𝐿22 − 𝐿21 𝐿12/𝐿11  Magnetic magnon conductance: 𝐺 ≡ 𝐿11 Thermomagnetic Effects Low temp.: ℏ/(2𝜏) ≪ 𝑘 𝐵 𝑇 ≪ 𝑔𝜇 𝐵 𝐵 Wiedemann-Franz Law for Magnon (𝜏:Magnon lifetime) Magnon (Boson) Electron (Fermion) `Universal’
  22. 22. e vs 𝝁 𝑩 Electron (metal) Magnon (FI) R. Franz and G. Wiedemann [Annalen der Physik 165, 497 (1853)] KN, P. Simon, and DL [Phys. Rev. B 92, 134425 (2015)] Fermion Boson WF law (Low temp.) 𝐿22 + 𝑂(𝜀 𝐹 −2 ) 𝐿11 ≡ 𝐾 𝜎 = 𝜋2 3 𝑘 𝐵 𝑒 2 𝑇 (Free electron at low temp.) 𝐿22 − 𝐿21 𝐿12 /𝐿11 𝐿11 ≡ 𝐾 𝐺 = 𝑘 𝐵 𝑔𝜇 𝐵 2 𝑇 [Low temp.: ℏ/(2𝜏) ≪ 𝑘 𝐵 𝑇 ≪ 𝑔𝜇 𝐵 𝐵] Lorenz number ℒ ≡ 𝜋2 3 𝑘 𝐵 𝒆 2 ℒm ≡ 𝑘 𝐵 𝒈𝝁 𝑩 2 Seebeck 𝑆 & Peltier Π 𝑆 ≡ 𝐿12 /𝐿11 , 𝛱 ≡ 𝐿21 /𝐿11 𝑆 = 𝐵/𝑇, 𝛱 = 𝐵 [Low temp.: ℏ/(2𝜏) ≪ 𝑘 𝐵 𝑇 ≪ 𝑔𝜇 𝐵 𝐵]  Universal Onsager relation 𝐿21 = 𝑇𝐿12 𝐿21 = 𝑇𝐿12 Thomson relation 𝛱 = 𝑇𝑆 𝛱 = 𝑇𝑆 Thermo-electric & –magnetic Effects
  23. 23. CONCLUSION Ratio of 𝐿𝑖𝑗 : 𝐾/𝐺, 𝑆, 𝛱  Universal thermomagnetic properties (i.e., Not depend on materials)  Each Onsager coefficient 𝐿𝑖𝑗: Depend on materials
  24. 24. SUMMARY 𝐾 𝐺 = 𝑘 𝐵 𝑔𝜇 𝐵 2 𝑇 ∝ 𝑇 𝐾 : Thermal magnon conductance, 𝐺: Magnetic magnon conductance Wiedemann-Franz Law for Magnon  Fundamental thermomagnetic relation of magnon transport in FI  Ratio of 𝐿𝑖𝑗: 𝐾/𝐺, 𝑆, 𝛱  Universal thermomagnetic properties Low temp.: ℏ/(2𝜏) ≪ 𝑘 𝐵 𝑇 ≪ 𝑔𝜇 𝐵 𝐵 𝑘B𝜇B `WF’ Magnet: 𝐺 Heat: 𝐾 Magnon (Boson) Electron (Fermion) `Universal’ Based on [Phys. Rev. B 92, 134425 (2015)] by KN, P. Simon, and D. Loss

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