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