magnonic wiedemann-franz law

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]