wiedemann-franz law for magnon transport
TRANSCRIPT
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 โKouki Nakataโ
MAIN MESSAGE
162 YEARS AGO
due to electron (Fermion)
[R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]
ใWiedemann-Franz Lawใ
๐2
3
๐B
๐
2
๐
Thermoelectric Effects in Metal
THEN
Thermomagnetic Effects in FI
QUESTION How expressed in `AN EQUATIONโ ?
due to magnon (Boson)
Universality
๐B
๐๐B
2
๐
ANSWER
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
WHY๏ผ We discuss from now on
BACKGROUND
Universal Thermomagnetic Relation of Magnon Transport
GOAL
FI๏ผLong-ranged magnetic order ``Magnon (spin-wave)โโ
๐B ๐B Magnet Heat
๏ผ
Universal Thermomagnetic Relation of Magnon Transport
Thermoelectric properties of Electron transport in metal
Wiedemann-Franz Law
Guiding principle
FI๏ผLong-ranged magnetic order ``Magnon (spin-wave)โโ
GOAL
Wiedemann-Franz Law [R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]
Thermoelectric properties of electron transport
Lorenz number โ โก๐2
3
๐๐ต
๐
2: Universal
๐พ
๐ =
๐2
3
๐๐ต
๐
2
๐
(๐พ: Thermal conductivity, ๐: Electrical conductivity)
Low temp.
๐๐
๐๐= ๐ฟ11 ๐ฟ12
๐ฟ21 ๐ฟ22๐ธ
๐ป๐
charge
Heat
Onsager matrix ๐ฟ๐๐
Thermoelectric Effects
Electron (metal) Magnon (FI)
WF law ๏ผLow temp.๏ผ
๐ฟ22 + ๐(๐๐นโ2)
๐ฟ11โ
๐พ
๐=
๐2
3
๐๐ต
๐
2
๐ ? Lorenz
number โ โก๐2
3
๐๐ต
๐
2
? Seebeck ๐ &
Peltier ฮ ๐ โก ๐ฟ12/๐ฟ11, ๐ฑ โก ๐ฟ21/๐ฟ11 Thomson relation: ๐ฑ = ๐๐ ?
Electron (metal) Magnon (FI)
WF law ๏ผLow temp.๏ผ
๐ฟ22 + ๐(๐๐นโ2)
๐ฟ11โ
๐พ
๐=
๐2
3
๐๐ต
๐
2
๐ ? Lorenz
number โ โก๐2
3
๐๐ต
๐
2
? Seebeck ๐ &
Peltier ฮ ๐ โก ๐ฟ12/๐ฟ11, ๐ฑ โก ๐ฟ21/๐ฟ11 Thomson relation: ๐ฑ = ๐๐ ?
๐ผm
๐ผ๐= ๐ฟ11 ๐ฟ12
๐ฟ21 ๐ฟ22๐ป๐ต๐ป๐
Magnet
Heat
Onsager matrix ๐ฟ๐๐
Thermomagnetic Effects
๐ผm
๐ผ๐= ๐ฟ11 ๐ฟ12
๐ฟ21 ๐ฟ22๐ป๐ต๐ป๐
WF
Magnet
Heat
Thermomagnetic Effects Onsager matrix ๐ฟ๐๐
Electron (metal) Magnon (FI)
WF law ๏ผLow temp.๏ผ
๐ฟ22 + ๐(๐๐นโ2)
๐ฟ11โ
๐พ
๐=
๐2
3
๐๐ต
๐
2
๐ ๐พ
๐บโก
๐ฟ22 โ ๐ฟ21๐ฟ12/๐ฟ11
๐ฟ11= ?
Lorenz number โ โก
๐2
3
๐๐ต
๐
2
โm = ?
Seebeck ๐ & Peltier ฮ
๐ โก ๐ฟ12/๐ฟ11, ๐ฑ โก ๐ฟ21/๐ฟ11 Thomson relation: ๐ฑ = ๐๐
What is their behaviors at low temp. ?
Charge
๐ Magnet
๐B
Heat
๐B
TARGET
Fermion VS Boson
``Wiedemann-Franz Lawโโ
[R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
Point
Thermal properties โ๐๐โ๏ผDifferent ? OR Universal ?
Magnon Wiedemann-Franz Law
Quantum-statistical properties are different
Electron ๐ = Fermion
Magnon ๐B = Boson
SYSTEM
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
Ferromagnetic Insulating Junction
๐ฝex โช ๐ฝ ๏ผweak coupling๏ผ
๐L
๐R
โ๐ต โก ๐ตR โ ๐ตL
โ๐ โก ๐R โ ๐L
Magnon currents Q. What happen when magnons are in condensation ? See [PRB 90, 144419 (2014)] & [PRB 92, 014422 (2015)]
Onsager matrix ๐ฟ๐๐
Magnetic current
Heat current
๐ฝex โช ๐ฝ,
( ๐: Lattice constant)
โ๐ต โก ๐ตR โ ๐ตL, โ๐ โก ๐R โ ๐L
๐R
๐L
Ferromagnetic Insulating Junction
๐ฟ11 โ ๐B2
๐ฟ22 โ ๐B2
๐ฟ12 โ ๐B๐B
๐ฟ21 โ ๐B๐B
RESULTS
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
Magnon Lorenz number: โm โก๐๐ต
๐๐๐ต
2: `Universalโ
๐พ
๐บ =
๐๐ต
๐๐๐ต
2
๐ โ ๐
Thermal magnon conductance: ๐พ โก ๐ฟ22 โ ๐ฟ21๐ฟ12/๐ฟ11
Magnetic magnon conductance: ๐บ โก ๐ฟ11
Thermomagnetic Effects
Low temp.๏ผ โ/(2๐) โช ๐๐ต๐ โช ๐๐๐ต๐ต
Wiedemann-Franz Law for Magnon (๐๏ผMagnon lifetime)
Magnon (Boson)
Electron (Fermion)
`Universalโ
e vs ๐๐ฉ 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
๏ผLow temp.๏ผ
๐ฟ22 + ๐(๐๐นโ2)
๐ฟ11โก
๐พ
๐=
๐2
3
๐๐ต
๐
2
๐
(Free electron at low temp.)
๐ฟ22 โ ๐ฟ21๐ฟ12/๐ฟ11
๐ฟ11โก
๐พ
๐บ=
๐๐ต
๐๐๐ต
2
๐
[Low temp.๏ผ โ/(2๐) โช ๐๐ต๐ โช ๐๐๐ต๐ต]
Lorenz number โ โก
๐2
3
๐๐ต
๐
2
โm โก๐๐ต
๐๐๐ฉ
2
Seebeck ๐ & Peltier ฮ
๐ โก ๐ฟ12/๐ฟ11, ๐ฑ โก ๐ฟ21/๐ฟ11 ๐ = ๐ต/๐, ๐ฑ = ๐ต [Low temp.๏ผ โ/(2๐) โช ๐๐ต๐ โช ๐๐๐ต๐ต]
Universal
Onsager relation
๐ฟ21 = ๐๐ฟ12 ๐ฟ21 = ๐๐ฟ12
Thomson relation
๐ฑ = ๐๐ ๐ฑ = ๐๐
Thermo-electric & โmagnetic Effects
CONCLUSION
Ratio of ๐ฟ๐๐: ๐พ/๐บ, ๐, ๐ฑ
Universal thermomagnetic properties (i.e., Not depend on materials)
Each Onsager coefficient ๐ฟ๐๐๏ผ Depend on materials
SUMMARY
๐พ
๐บ=
๐๐ต
๐๐๐ต
2
๐ โ ๐
๐พ : Thermal magnon conductance, ๐บ: Magnetic magnon conductance
Wiedemann-Franz Law for Magnon
Fundamental thermomagnetic relation of magnon transport in FI
Ratio of ๐ฟ๐๐: ๐พ/๐บ, ๐, ๐ฑ Universal thermomagnetic properties
Low temp.๏ผ โ/(2๐) โช ๐๐ต๐ โช ๐๐๐ต๐ต
๐B ๐B ฬ WFโ
Magnet: ๐บ Heat: ๐พ
Magnon (Boson)
Electron (Fermion)
`Universalโ
Based on [Phys. Rev. B 92, 134425 (2015)] by KN, P. Simon, and D. Loss