The document discusses the Wiedemann-Franz law for magnon transport in ferromagnetic insulators. It shows that at low temperatures, the ratio of the thermal magnon conductance K to the magnetic magnon conductance G, as well as other Onsager coefficients, are universal and do not depend on material properties. Specifically, the ratio K/G is equal to kB/gμB2T, analogous to the Wiedemann-Franz law for electrons in metals. This establishes a fundamental thermomagnetic relation for magnon transport in ferromagnetic insulators.

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.  MAIN MESSAGE

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. THEN

5. Thermomagnetic Effects in FI
QUESTION
 How expressed in `AN EQUATION’ ?
due to magnon (Boson)
Universality

6. 𝑘B
𝑔𝜇B
2
𝑇
ANSWER
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]

7. WHY？ We discuss from now on

8.  BACKGROUND

9. Universal Thermomagnetic Relation
of Magnon Transport
GOAL
 FI：Long-ranged magnetic order  ``Magnon (spin-wave)’’
𝑘B
𝜇B
Magnet Heat
？

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. 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. 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. 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. 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. 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. Point
 Thermal properties “𝒌 𝐁”：Different ? OR Universal ?
Magnon Wiedemann-Franz Law
 Quantum-statistical properties are different
Electron 𝒆 = Fermion
Magnon 𝜇B = Boson

17.  SYSTEM
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]

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. 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.  RESULTS
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]

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. 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. 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. 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