magnonic wiedemann-franz law
TRANSCRIPT
Kouki Nakata
University of Basel
All the responsibilities of this slide rest with Kouki Nakata (Dec. 2016) Enjoy also talk by DL : https://www.youtube.com/watch?v=56T4CmjkA5c&list=PLS3nw8GL8hAXeJFCXcziJMHB1Yb_HLncd
Magnonic Wiedemann-Franz Law
KN, P. Simon, and D. Loss: Phys. Rev. B 92, 134425 (2015) See also [KN, J. Klinovaja & D. Loss (2016), arXiv:1611.09752] & review article [KN, P. Simon & D. Loss, arXiv:1610.08901]
Magnon Carries 𝜇B & 𝑘B
≤ ≪
Magnon 𝜇B 𝑘B
Low-energy collective mode in insulating magnet
Yes !
QUESTION
Can magnon 𝜇B (boson) transport be similar to electron 𝑒 (fermion) transport ?
Electron 𝑒 = Fermion
Magnon 𝜇B = Boson
Wiedemann-Franz (WF) law Franz and Wiedemann, Annalen der Physik (1853)
Magnonic Wiedemann-Franz law KN, P. Simon & D. Loss, PRB (2015)
Superconductors
Onnes (1911)
Quasi-equilibrium magnon condensate Demokritov et al., Nature (2006)
Condensed magnon current Hillebrands-group, Nat. Phys. (2016)
Josephson effect Josephson, Phys. Lett. (1962)
Magnonic Josephson effect KN, K. A. van Hoogdalem, P. Simon & D. Loss, PRB (2014)
KN, P. Simon & D. Loss, PRB (2015)
Quantum Hall effetc (QHE) Klitzing et al., PRL (1980)
Magnonic QHE KN, J. Klinovaja & D. Loss (2016), arXiv:1611.09752
QUESTION
Can magnon 𝜇B (boson) transport be similar to electron 𝑒 (fermion) transport ?
See review article [KN, P. Simon & D. Loss, arXiv:1610.08901]
Universal Thermomagnetic Relation of Magnon Transport
?
GOAL
for electron transport in metals
Wiedemann-Franz Law
Y. Kajiwara et al., Nature 464, 262 (2010)
Wiedemann-Franz Law in Metals Franz and Wiedemann, Annalen der Physik 165, 497 (1853)
Universal Lorenz number:
At low temperature 𝑇 ≪ 𝜖F/𝑘B~104 K
𝐾: Thermal conductivity 𝜎: Electrical conductivity
Thermoelectrics in Metal
Seebeck & Peltier
WF law (Low temp.)
?
?
Electron = Fermion Magnon = Boson Textbook by Ashcroft & Mermin
Lorenz number =
= =
=
? ?
−
Charge
Heat
−
Charge
Heat
K
Thermal Conductivity 𝐾 ≈ 𝐿22 for Fermions
Seebeck & Peltier
WF law (Low temp.)
?
Electron = Fermion Magnon = Boson Textbook by Ashcroft & Mermin
Lorenz number =
= =
=
? ?
?
*Lifshitz & Pitaevskii (Vol. 10)
*
−
Magnon
Thermal Conductivity 𝐾 ≠ 𝐿22 for Magnons KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
Seebeck & Peltier
WF law (Low temp.)
?
?
Electron = Fermion Magnon = Boson Textbook by Ashcroft & Mermin
Lorenz number =
= =
=
K
? ?
*
*Lifshitz & Pitaevskii (Vol. 10)
Heat
Thermal Conductivity 𝐾 ≠ 𝐿22 for Magnons
Textbook by Ashcroft & Mermin Eq. (13.56): K is measured under conditions of no quasi-particle current
𝐈m = 𝐿11𝜵𝐵 −𝐿12𝜵𝑇 = 0 𝜵𝐵∗ =𝐿12
𝐿11𝜵𝑇
!
𝐈𝑄 = 𝐿21𝜵𝐵∗− 𝐿22𝜵𝑇 = −(𝐿22− 𝐿21𝐿12/𝐿11)𝜵𝑇
Thermal conductivity 𝐾: 𝐈𝑄 ≡ −𝐾 ∙ 𝜵𝑇 with !
𝐈m = 0
K
Magnetization gradient
Johnson & Silsbee (1987) Basso et al. (2016)
−
Magnon K
Heat
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
Thermomagnetics in Ferromagnetic Insulator (FI)
Seebeck & Peltier
WF law (Low temp.)
?
?
Electron = Fermion Magnon = Boson Textbook by Ashcroft & Mermin
Lorenz number =
= =
=
? ?
*
*Lifshitz & Pitaevskii (Vol. 10)
−
Magnon
Heat
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
−
Magnon
Heat
Thermomagnetics in Ferromagnetic Insulator (FI)
Magnon Seebeck 𝒮
Seebeck & Peltier
WF law (Low temp.)
?
?
Electron = Fermion Magnon = Boson Textbook by Ashcroft & Mermin
Lorenz number =
= =
=
? ?
*
*Lifshitz & Pitaevskii (Vol. 10)
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
−
Magnon
Heat
Magnon Peltier: Π
Thermomagnetics in Ferromagnetic Insulator (FI)
Seebeck & Peltier
WF law (Low temp.)
?
?
Electron = Fermion Magnon = Boson Textbook by Ashcroft & Mermin
Lorenz number =
= =
=
? ?
*
*Lifshitz & Pitaevskii (Vol. 10)
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
−
Magnon
Heat
Onsager relation
Thermomagnetics in Ferromagnetic Insulator (FI)
Seebeck & Peltier
WF law (Low temp.)
?
?
Electron = Fermion Magnon = Boson Textbook by Ashcroft & Mermin
Lorenz number =
= =
=
? ?
*
*Lifshitz & Pitaevskii (Vol. 10)
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
−
Magnon
Heat
Thermomagnetics in Ferromagnetic Insulator (FI)
Seebeck & Peltier
WF law (Low temp.)
?
?
Electron = Fermion Magnon = Boson Textbook by Ashcroft & Mermin
Lorenz number =
= =
=
WF
? ?
*
*Lifshitz & Pitaevskii (Vol. 10)
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
QUESTION
Magnonic Wiedemann-Franz Law
Magnon 𝜇B : Boson
Electron 𝑒 : Fermion
Bose-Einstein statistics vs Fermi-Dirac statistics
Specific heat of electrons:
𝒞el ∝ 𝑇 Specific heat of phonons:
𝒞ph ∝ 𝑇3
WELL-KNOWN: Qualitatively different behavior at low temperature
Q. Still, Linear-in-𝑇 behavior for bosons ? Universal ?
Magnon current:
Heat current:
Onsager matrix 𝐿𝑖𝑗 :
System: Ferromagnetic Insulating Junction
Driving forces: ≪ 𝐵 ≪ 𝑇
𝜔𝑘L(R) = 2𝐽𝑆𝑎2𝑘2+𝑔𝜇𝐵𝐵L(R)
Tunneling:
Weak coupling :
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
Magnon & Heat Currents
Magnon current
Heat current
𝜏: Magnon lifetime: Phenomenologically introduced
Linear response
Bose function
Onsager relation:
Onsager Matrix 𝐿𝑖𝑗
Polylogarithm function:
Exponential integral:
Euler constant: Cross-section area of the junction interface:
𝐿𝑖𝑗 depends on material
Onsager relation
Magnon current
Heat current
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
Onsager Matrix 𝐿𝑖𝑗
Polylogarithm function:
Exponential integral:
Euler constant: Cross-section area of the junction interface:
𝐿𝑖𝑗 depends on material
Onsager relation
Magnon current
Heat current
NOTE: 𝑏 𝑇 ≡𝑔𝜇B𝐵
𝑘B𝑇↗ at low temperature
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
𝑔𝜇B𝐵
𝑘B𝑇
Magnetic conductance 𝐺:
: For fermions
Thermal conductance 𝐾 for magnons (bosons)
Magnon WF law
Wiedemann-Franz Law for Magnons
Magnon current
Heat current
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
Wiedemann-Franz Law for Magnons
Low temp.:
Magnon Lorenz number: UNIVERSAL
Independent of materials
e.g., 𝜏 = 100ns, 𝑇 ≤ 1K & 𝐵 = 5T
(If omitted the off-diagonal 𝐿21 & 𝐿12 The WF law is not reproduced)
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
Note: 1) We checked that 3- and 4-magnon processes are negligible at low temp.
KN et al., PRB 92, 134425 (2015)
2) At 𝑇 = 𝒪(10−1) K, phonon contributions are negligible
Adachi et al., APL 97, 252506 (2010)
3) The WF law holds in dipole-dipole int. (YIG)
Wiedemann-Franz Law for Magnons
e.g., 𝜏 = 100ns, 𝑇 ≤ 1K & 𝐵 = 5T
(If omitted the off-diagonal 𝐿21 & 𝐿12 The WF law is not reproduced)
Low temp.:
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
VS
(Free electron at low temp.) Low temp.:
Electron (metal) Magnon (FI)
R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)
KN, Simon, and Loss, Phys. Rev. B 92, 134425 (2015)
Fermion Boson Statistics
Lorenz number
WF law (Low temp.)
SUMMARY
Onsager-Thomson relation
Seebeck & Peltier
Ratios of 𝐿𝑖𝑗 , WF law, Seebeck, and Peltier coefficients are material independent
Magnonic Wiedemann-Franz Law: KN, P. Simon, and D. Loss, Phys. Rev. B 92, 134425 (2015) See also [KN, J. Klinovaja & D. Loss (2016), arXiv:1611.09752] & review article [KN, P. Simon & D. Loss, arXiv:1610.08901]