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 βKouki Nakataβ
MAIN MESSAGE
162 YEARS AGO
due to electron (Fermion)
[R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]
γWiedemann-Franz Lawγ
π2
3
πB
π
2
π
Thermoelectric Effects in Metal
THEN
Thermomagnetic Effects in FI
QUESTION How expressed in `AN EQUATIONβ ?
due to magnon (Boson)
Universality
πB
ππB
2
π
ANSWER
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
WHYοΌ We discuss from now on
BACKGROUND
Universal Thermomagnetic Relation of Magnon Transport
GOAL
FIοΌLong-ranged magnetic order ``Magnon (spin-wave)ββ
πB πB Magnet Heat
οΌ
Universal Thermomagnetic Relation of Magnon Transport
Thermoelectric properties of Electron transport in metal
Wiedemann-Franz Law
Guiding principle
FIοΌLong-ranged magnetic order ``Magnon (spin-wave)ββ
GOAL
Wiedemann-Franz Law [R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]
Thermoelectric properties of electron transport
Lorenz number β β‘π2
3
ππ΅
π
2: Universal
πΎ
π =
π2
3
ππ΅
π
2
π
(πΎ: Thermal conductivity, π: Electrical conductivity)
Low temp.
ππ
ππ= πΏ11 πΏ12
πΏ21 πΏ22πΈ
π»π
charge
Heat
Onsager matrix πΏππ
Thermoelectric Effects
Electron (metal) Magnon (FI)
WF law οΌLow temp.οΌ
πΏ22 + π(ππΉβ2)
πΏ11β
πΎ
π=
π2
3
ππ΅
π
2
π ? Lorenz
number β β‘π2
3
ππ΅
π
2
? Seebeck π &
Peltier Ξ π β‘ πΏ12/πΏ11, π± β‘ πΏ21/πΏ11 Thomson relation: π± = ππ ?
Electron (metal) Magnon (FI)
WF law οΌLow temp.οΌ
πΏ22 + π(ππΉβ2)
πΏ11β
πΎ
π=
π2
3
ππ΅
π
2
π ? Lorenz
number β β‘π2
3
ππ΅
π
2
? Seebeck π &
Peltier Ξ π β‘ πΏ12/πΏ11, π± β‘ πΏ21/πΏ11 Thomson relation: π± = ππ ?
πΌm
πΌπ= πΏ11 πΏ12
πΏ21 πΏ22π»π΅π»π
Magnet
Heat
Onsager matrix πΏππ
Thermomagnetic Effects
πΌm
πΌπ= πΏ11 πΏ12
πΏ21 πΏ22π»π΅π»π
WF
Magnet
Heat
Thermomagnetic Effects Onsager matrix πΏππ
Electron (metal) Magnon (FI)
WF law οΌLow temp.οΌ
πΏ22 + π(ππΉβ2)
πΏ11β
πΎ
π=
π2
3
ππ΅
π
2
π πΎ
πΊβ‘
πΏ22 β πΏ21πΏ12/πΏ11
πΏ11= ?
Lorenz number β β‘
π2
3
ππ΅
π
2
βm = ?
Seebeck π & Peltier Ξ
π β‘ πΏ12/πΏ11, π± β‘ πΏ21/πΏ11 Thomson relation: π± = ππ
What is their behaviors at low temp. ?
Charge
π Magnet
πB
Heat
πB
TARGET
Fermion VS Boson
``Wiedemann-Franz Lawββ
[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 βππβοΌDifferent ? OR Universal ?
Magnon Wiedemann-Franz Law
Quantum-statistical properties are different
Electron π = Fermion
Magnon πB = Boson
SYSTEM
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
Ferromagnetic Insulating Junction
π½ex βͺ π½ οΌweak couplingοΌ
πL
πR
βπ΅ β‘ π΅R β π΅L
βπ β‘ πR β πL
Magnon currents Q. What happen when magnons are in condensation ? See [PRB 90, 144419 (2014)] & [PRB 92, 014422 (2015)]
Onsager matrix πΏππ
Magnetic current
Heat current
π½ex βͺ π½,
( π: Lattice constant)
βπ΅ β‘ π΅R β π΅L, βπ β‘ πR β πL
πR
πL
Ferromagnetic Insulating Junction
πΏ11 β πB2
πΏ22 β πB2
πΏ12 β πBπB
πΏ21 β πBπB
RESULTS
[KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015)]
Magnon Lorenz number: βm β‘ππ΅
πππ΅
2: `Universalβ
πΎ
πΊ =
ππ΅
πππ΅
2
π β π
Thermal magnon conductance: πΎ β‘ πΏ22 β πΏ21πΏ12/πΏ11
Magnetic magnon conductance: πΊ β‘ πΏ11
Thermomagnetic Effects
Low temp.οΌ β/(2π) βͺ ππ΅π βͺ πππ΅π΅
Wiedemann-Franz Law for Magnon (ποΌMagnon lifetime)
Magnon (Boson)
Electron (Fermion)
`Universalβ
e vs ππ© 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
οΌLow temp.οΌ
πΏ22 + π(ππΉβ2)
πΏ11β‘
πΎ
π=
π2
3
ππ΅
π
2
π
(Free electron at low temp.)
πΏ22 β πΏ21πΏ12/πΏ11
πΏ11β‘
πΎ
πΊ=
ππ΅
πππ΅
2
π
[Low temp.οΌ β/(2π) βͺ ππ΅π βͺ πππ΅π΅]
Lorenz number β β‘
π2
3
ππ΅
π
2
βm β‘ππ΅
πππ©
2
Seebeck π & Peltier Ξ
π β‘ πΏ12/πΏ11, π± β‘ πΏ21/πΏ11 π = π΅/π, π± = π΅ [Low temp.οΌ β/(2π) βͺ ππ΅π βͺ πππ΅π΅]
Universal
Onsager relation
πΏ21 = ππΏ12 πΏ21 = ππΏ12
Thomson relation
π± = ππ π± = ππ
Thermo-electric & βmagnetic Effects
CONCLUSION
Ratio of πΏππ: πΎ/πΊ, π, π±
Universal thermomagnetic properties (i.e., Not depend on materials)
Each Onsager coefficient πΏπποΌ Depend on materials
SUMMARY
πΎ
πΊ=
ππ΅
πππ΅
2
π β π
πΎ : Thermal magnon conductance, πΊ: Magnetic magnon conductance
Wiedemann-Franz Law for Magnon
Fundamental thermomagnetic relation of magnon transport in FI
Ratio of πΏππ: πΎ/πΊ, π, π± Universal thermomagnetic properties
Low temp.οΌ β/(2π) βͺ ππ΅π βͺ πππ΅π΅
πB πB Μ WFβ
Magnet: πΊ Heat: πΎ
Magnon (Boson)
Electron (Fermion)
`Universalβ
Based on [Phys. Rev. B 92, 134425 (2015)] by KN, P. Simon, and D. Loss
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