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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”
MAIN MESSAGE
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
THEN
Thermomagnetic Effects in FI
QUESTION How expressed in `AN EQUATION’ ?
due to magnon (Boson)
Universality
𝑘B
𝑔𝜇B
2
𝑇
ANSWER
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
WHY? We discuss from now on
BACKGROUND
Universal Thermomagnetic Relation of Magnon Transport
GOAL
FI:Long-ranged magnetic order ``Magnon (spin-wave)’’
𝑘B 𝜇B Magnet Heat
?
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
Wiedemann-Franz Law [R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]
Thermoelectric properties of electron transport
Lorenz number ℒ ≡𝜋2
3
𝑘𝐵
𝑒
2: Universal
𝐾
𝜎 =
𝜋2
3
𝑘𝐵
𝑒
2
𝑇
(𝐾: Thermal conductivity, 𝜎: Electrical conductivity)
Low temp.
𝑗𝑒
𝑗𝑄= 𝐿11 𝐿12
𝐿21 𝐿22𝐸
𝛻𝑇
charge
Heat
Onsager matrix 𝐿𝑖𝑗
Thermoelectric Effects
Electron (metal) Magnon (FI)
WF law (Low temp.)
𝐿22 + 𝑂(𝜀𝐹−2)
𝐿11≈
𝐾
𝜎=
𝜋2
3
𝑘𝐵
𝑒
2
𝑇 ? Lorenz
number ℒ ≡𝜋2
3
𝑘𝐵
𝑒
2
? Seebeck 𝑆 &
Peltier Π 𝑆 ≡ 𝐿12/𝐿11, 𝛱 ≡ 𝐿21/𝐿11 Thomson relation: 𝛱 = 𝑇𝑆 ?
Electron (metal) Magnon (FI)
WF law (Low temp.)
𝐿22 + 𝑂(𝜀𝐹−2)
𝐿11≈
𝐾
𝜎=
𝜋2
3
𝑘𝐵
𝑒
2
𝑇 ? Lorenz
number ℒ ≡𝜋2
3
𝑘𝐵
𝑒
2
? Seebeck 𝑆 &
Peltier Π 𝑆 ≡ 𝐿12/𝐿11, 𝛱 ≡ 𝐿21/𝐿11 Thomson relation: 𝛱 = 𝑇𝑆 ?
𝐼m
𝐼𝑄= 𝐿11 𝐿12
𝐿21 𝐿22𝛻𝐵𝛻𝑇
Magnet
Heat
Onsager matrix 𝐿𝑖𝑗
Thermomagnetic Effects
𝐼m
𝐼𝑄= 𝐿11 𝐿12
𝐿21 𝐿22𝛻𝐵𝛻𝑇
WF
Magnet
Heat
Thermomagnetic Effects Onsager matrix 𝐿𝑖𝑗
Electron (metal) Magnon (FI)
WF law (Low temp.)
𝐿22 + 𝑂(𝜀𝐹−2)
𝐿11≈
𝐾
𝜎=
𝜋2
3
𝑘𝐵
𝑒
2
𝑇 𝐾
𝐺≡
𝐿22 − 𝐿21𝐿12/𝐿11
𝐿11= ?
Lorenz number ℒ ≡
𝜋2
3
𝑘𝐵
𝑒
2
ℒm = ?
Seebeck 𝑆 & Peltier Π
𝑆 ≡ 𝐿12/𝐿11, 𝛱 ≡ 𝐿21/𝐿11 Thomson relation: 𝛱 = 𝑇𝑆
What is their behaviors at low temp. ?
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)]
Point
Thermal properties “𝒌𝐁”:Different ? OR Universal ?
Magnon Wiedemann-Franz Law
Quantum-statistical properties are different
Electron 𝒆 = Fermion
Magnon 𝜇B = Boson
SYSTEM
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
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)]
Onsager matrix 𝐿𝑖𝑗
Magnetic current
Heat current
𝐽ex ≪ 𝐽,
( 𝑎: Lattice constant)
∆𝐵 ≡ 𝐵R − 𝐵L, ∆𝑇 ≡ 𝑇R − 𝑇L
𝑇R
𝑇L
Ferromagnetic Insulating Junction
𝐿11 ∝ 𝜇B2
𝐿22 ∝ 𝑘B2
𝐿12 ∝ 𝜇B𝑘B
𝐿21 ∝ 𝜇B𝑘B
RESULTS
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
Magnon Lorenz number: ℒm ≡𝑘𝐵
𝑔𝜇𝐵
2: `Universal’
𝐾
𝐺 =
𝑘𝐵
𝑔𝜇𝐵
2
𝑇 ∝ 𝑇
Thermal magnon conductance: 𝐾 ≡ 𝐿22 − 𝐿21𝐿12/𝐿11
Magnetic magnon conductance: 𝐺 ≡ 𝐿11
Thermomagnetic Effects
Low temp.: ℏ/(2𝜏) ≪ 𝑘𝐵𝑇 ≪ 𝑔𝜇𝐵𝐵
Wiedemann-Franz Law for Magnon (𝜏:Magnon lifetime)
Magnon (Boson)
Electron (Fermion)
`Universal’
e vs 𝝁𝑩 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(𝜀𝐹−2)
𝐿11≡
𝐾
𝜎=
𝜋2
3
𝑘𝐵
𝑒
2
𝑇
(Free electron at low temp.)
𝐿22 − 𝐿21𝐿12/𝐿11
𝐿11≡
𝐾
𝐺=
𝑘𝐵
𝑔𝜇𝐵
2
𝑇
[Low temp.: ℏ/(2𝜏) ≪ 𝑘𝐵𝑇 ≪ 𝑔𝜇𝐵𝐵]
Lorenz number ℒ ≡
𝜋2
3
𝑘𝐵
𝒆
2
ℒm ≡𝑘𝐵
𝒈𝝁𝑩
2
Seebeck 𝑆 & Peltier Π
𝑆 ≡ 𝐿12/𝐿11, 𝛱 ≡ 𝐿21/𝐿11 𝑆 = 𝐵/𝑇, 𝛱 = 𝐵 [Low temp.: ℏ/(2𝜏) ≪ 𝑘𝐵𝑇 ≪ 𝑔𝜇𝐵𝐵]
Universal
Onsager relation
𝐿21 = 𝑇𝐿12 𝐿21 = 𝑇𝐿12
Thomson relation
𝛱 = 𝑇𝑆 𝛱 = 𝑇𝑆
Thermo-electric & –magnetic Effects
CONCLUSION
Ratio of 𝐿𝑖𝑗: 𝐾/𝐺, 𝑆, 𝛱
Universal thermomagnetic properties (i.e., Not depend on materials)
Each Onsager coefficient 𝐿𝑖𝑗: Depend on materials
SUMMARY
𝐾
𝐺=
𝑘𝐵
𝑔𝜇𝐵
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) ≪ 𝑘𝐵𝑇 ≪ 𝑔𝜇𝐵𝐵
𝑘B 𝜇B ̀ 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