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Kouki Nakata
University of Basel Switzerland
Magnon Transport Theory
All the responsibility of this slide rests with `Kouki Nakata' (2016)
10 pages digest of the works at Basel 2014-2015
Yes !
[Phys. Rev. B 90, 144419 (2014)] [Phys. Rev. B 92, 014422 (2015)] [Phys. Rev. B 92, 134425 (2015)]
We have established it: Magnon counterpart of electron transport
Q. Can we control magnon transport 𝝁𝑩 like electrons 𝒆 ?
FINAL GOAL Establish valid methods to control magnon transport 𝝁𝐁
Charge transport 𝑒 Magnon transport 𝜇B
Wiedemann-Franz (WF) law [R. Franz and G. Wiedemann,
Annalen der Physik 165, 497 (1853)]
Thermoelectric property
Magnon Wiedemann-Franz law [K. Nakata et al., Phys. Rev. B 92, 134425 (2015)]
Thermomagnetic property
Superconducting state [H. K. Onnes (1911)]
Persistent charge current [M. Buttiker et al. Phys. Lett. A, 96, 365 (1983)]
Magnon-BEC [S. O. Demokritov et al., Nature 443, 430 (2006)]
Persistent magnon-BEC current [K. Nakata et al., Phys. Rev. B 90, 144419 (2014)]
Josephson effect [B. D. Josephson, Phys. Lett. 1, 251 (1962)]
Magnon Josephson effect [K. Nakata et al., Phys. Rev. B 90, 144419 (2014)]
Quantum Hall effect [K. v. Klitzing et al., Phys. Rev. Lett. 45, 494 (1980)]
Magnon quantum Hall effect [K. Nakata & D. Loss, to be submitted (2016)]
Find the counterpart !!
Guiding Principle
Magnon Wiedemann-Franz Law
VS
(Free electron at low temp.) Low temp.:
Electron (metal) Magnon (FI)
R. Franz and G. Wiedemann [Annalen der Physik 165, 497 (1853)]
K. Nakata and D. Loss [Phys. Rev. B 92, 134425 (2015)]
Fermion Boson Statistics
Lorenz number
WF law (Low temp.)
𝑇-linear behavior = Universal
1853
[R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]
Wiedemann-Franz Law for Electron Transport in Metal
Wiedemann-Franz Law for Magnon Transport in FI [K. Nakata and D. Loss, Phys. Rev. B 92, 134425 (2015)]
2015
Experiment [S. O. Demokritov et al., Nature 443, 430 (2006)]
Quasi-equilibrium Magnon-BEC
Experimental result by [A. A. Serga et al., Nat. commun. 5, 3452 (2014)]
Microwave pumping: Room temperature
Magnon VS Magnon-BEC
Incoherent spin precession Macroscopic coherent spin precession Macroscopic spins
= Sum of variety kinds of modes Macroscopic number of magnons occupies a single state
Quasi-equilibrium condensation
Part I: Magnon Part II: Magnon-BEC
= Spin-wave ~ Superconducting state of spin-wave
dc ac : Josephson effect
BEC
BEC
Magnetic field difference:
Part II: Condensed magnon (BEC) Part I: Non-condensed magnon
ac/dc Properties
J J J
J
J
J
J J
BEC
Magnon-BEC Ring Analogous to superconducting ring
A-C phase Persistent magnon-BEC current
E (A-C phase)
(Note; as long as magnons are in condensation)
Non-condensed magnon
Cylindrical wire
Condensed magnon
(1) Magnetic current (2) (3)
𝜇B
𝑉m 𝐼m
Electromagnetism by Magnon Current
[D. Loss and P. M. Goldbart, Phys. Lett. A 215, 197 (1996)] [F. Meier and D. Loss, Phys. Rev. Lett. 90, 167204 (2003)]
(Flow of magnetic dipole)
𝐸m
Magnon Transport Ferromagnetic Insulator
Universal thermomagnetic relation Magnon Seebeck & Peltier effects
III. Measurement II. Magnon-BEC Berry phase Josephson & persistent currents
Electromagnetic control Direct detection
I. Wiedemann-Franz Law for Magnon in FI
SUMMARY
[Phys. Rev. B 90, 144419 (2014)] [Phys. Rev. B 92, 014422 (2015)] [Phys. Rev. B 92, 134425 (2015)]
Supplemental Material
Appendix: Part I
Magnon & Heat Currents
Magnon current
Heat current
: Magnon lifetime (phenomenologically introduced)
Onsager relation:
Integrating over
Linear response:
Magnon & Heat Currents
Onsager Matrix Magnon current
Heat current
Onsager coefficient
Onsager relation
Polylogarithm function:
Exponential integral: Euler constant:
Cross-section area of the junction interface:
Thermal Conductance 𝑲 for Boson
Note: Definition of thermal conductance
with
WF law
Magnon current Heat current
Magnetic conductance: 𝑮
Thermal conductance: 𝑲
for fermions
for bosons (magnons)
Thermo-electric & –magnetic Effects VS
(Free electron at low temp.) Low temp.:
Electron (metal) Magnon (FI)
R. Franz and G. Wiedemann [Annalen der Physik 165, 497 (1853)]
K. Nakata, P. Simon, and D. Loss [Phys. Rev. B 92, 134425 (2015)]
Fermion Boson Statistics
Onsager relation
Thomson relation
Seebeck & Peltier
Lorenz number
WF law (Low temp.)
REMARK There was a possibility at low temp.:
Magnon WF law in FI:
𝐾
𝐺= (
𝑘B
𝑔𝜇B)2𝑇 ∙
𝑘B𝑇
𝑔𝜇B𝐵
𝑛−1
∝ 𝑇𝑛
𝐾
𝐺= (
𝑘B
𝑔𝜇B)2𝑇 ∝ 𝑇
Anisotropic spin 𝜂 ≠ 1 Magnon-magnon interactions
At such low temperatures: Phonon contributions are negligibly small
WF law & Onsager relations: Broken
Contributions of the breakings: Negligibly small at low temperatures 𝒪(10−1)K
WF law & Onsager relations: Approximately satisfied at such low temperatures [Note: Originally (𝜂 = 1), the WF law is realized at such low temperatures]
Broken Relations & Low Temperature
[H. Adachi et al., Appl. Phys. Lett. 97, 252506 (2010)]
Magnon VS Magnon-BEC
Incoherent spin precession Macroscopic coherent spin precession = Macroscopic spins
= Sum of variety kinds of modes = Macroscopic number of magnons occupies a single state
Quasi-equilibrium condensation
Number density: Number density:
Part I: Magnon Part II: Magnon-BEC
= Spin-wave ~ Superconducting state of spin-wave
Cooper pair: 𝑐𝒌↑𝑐−𝒌↓ BCS ≠ 0
Appendix: Part II
Microwave pumping Non-equilibrium steady state
Quasi-equilibrium magnon-BEC = [Metastable state]
≠ [Ground state]
Thermalization
Microwave: Switched off
FMR
B B
Quasi-equilibrium magnon-BEC
[C. D. Batista et al., Rev. Mod. Phys., 86, 563 (2014)]
= Dynamical condensation ≠ Thermal condensation
U(1)-symmetry: Broken U(1)-symmetry: Recovered
Quasi-equilibrium Magnon-BEC
Quasi-equilibrium Magnon-BEC [C. D. Batista et al., Rev. Mod. Phys., 86, 563 (2014)]
= Dynamical condensation ≠ Thermal condensation
Magnon-BEC Order Parameter [Textbook by Leggett] BEC: Einstein for free particles (i.e., no interactions)
Single-particle density matrix
𝜌1(𝒓, 𝒓′; 𝑡);
Probability amplitude 𝜌1(𝒓, 𝒓′; 𝑡) ≡ 𝜓 +(𝒓𝑡)𝜓(𝒓′𝑡)
(𝜓: Bose field)
Single eigenvalue Single BEC Several eigenvalues Fragmented BEC
Penrose & Onsager (1956)
lim𝒓−𝒓′→∞
𝜌1 𝒓, 𝒓′; 𝑡 = Ψ∗ 𝒓𝑡 Ψ (𝒓′𝑡)
Ψ(𝒓𝑡) ≡ 𝜓 (𝒓𝑡) : BEC order parameter = Off-diagonal long-range order (ODLRO) Widely used in BEC community
Yang (1962)
Extension of definition including interactions
Quasi-equilibrium magnon-BEC by microwave pumping satisfies this condition Experiment [S. O. Demokritov et al., Nature 443, 430 (2006)]
Quantum ? OR Classical ? [A. Ruckriegel and P. Kopietz, PRL 115, 157203 (2015)]
Magnetic field difference:
Period of ac Josephson effect:
Parameter values:
Josephson current:
Josephson magnon current:
Adjusting parameters:
10 ns
ac Josephson Effect
ac Josephson Effect: Nonlinear Effect
Magnon Josephson Eq.:
Josephson current:
Time-evolution of phase:
Nonlinear effect: 𝑧(0) ≠ 0 & Δ𝐵 = 0 ac Josephson effect Period 𝑇~10ns at weak 𝐽ex
Nonlinear effect
Linear effect
Experimental reach 1T/cm: Linear effect ≪ Nonlinear effect
Period 𝑻 of ac Josephson effect:
e.g.:
Within experimental reach
𝑧 0 = 0.6
Initial population imbalance:
dc Josephson Effect
Electric field Magnetic field
No A-C phase A-C phase
Electromagnetically realizable by applying an increasing magnetic field:
: dc effect ? : dc effect
Macroscopic Quantum Self-Trapping
MQST
(a)
(b)-(d)
MQST occurs when
``Direct Observation of Tunneling and Nonlinear Self-Trapping in a Single Bosonic Josephson Junction’’ [M. Albiez et al., PRL 95, 010402 (2005)]
MQST in Cold Atoms Already experimentally observed [M. Albiez et al., PRL 95, 010402 (2005)]
time (p = 50)
Destabilized deviation: 1/p << 1
Stable
Magnon-BEC Ring
Quantization:
time
1/p
1
Device for Direct Measurement To detect persistent quantized magnon-BEC current in the ring
Appendix: Others
3-dim Cubic Ferromagnet
Holstein-Primakoff (H-P) transformation
Heisenberg spin model:
Standard textbook [K. Kubo] on magnetism tells us:
Fourier transformation:
Parabolic dispersion: → 0 (𝑘 → 0)
Magnon = A kind of Nambu-Goldstone mode
= Massless particle = Non-relativistic magnon
𝜔𝑘
𝑘
Picture from google search
d-dim Cubic Anti-Ferromagnet (d≥3)
Bogoliubov transformation New Magnon Operators: 𝛼 & 𝛽
Diagonalization:
Linear dispersion: Relativistic magnon = Dirac magnon on d-dim AF
𝑘
𝜔𝑘
Standard textbook [K. Kubo] on magnetism tells us:
Nambu-Goldstone (NG) Theorem Magnon = A kind of NG mode (particle)
A continuous symmetry is spontaneously broken (SSB) Massless particles = NG boson
Heisenberg model: SSB of SO(3) Magnon = NG mode
B
A
A B
Magnon
[TEXTBOOK by Peskin]
``Rough & intuitive’’ correspondence
(massless)
Picture from wiki.
Picture from Google search
The relation between [# of broken symmetries] & [# of NG particles]: See [Watanabe-Murayama] & [Hidaka]
Mermin–Wagner–Hohenberg-Coleman theorem:
Continuous symmetries cannot be spontaneously broken (NO SSB):
- 𝑑 ≤ 2 - At finite temperature - Sufficiently short-range interactions
Absence of NG particles (e.g. magnons) on 𝑑 ≤ 2
Why 3-dim ?
See also recent development: [Phys. Rev. Lett. 107, 107201 (2011)] D. Loss, F. L. Pedrocchi, and A. J. Leggett
NOTE: The absence of SSB is valid only in the thermodynamic limit Ordering in finite size at finite temperatures is possible
Picture from Google search
Lattice structure
Band structure
Dirac Magnon on 2-dim Honeycomb Lattice
Magnon Dirac Eq.
[arXiv:1512.04902] J. Fransson, A. M. Black-Schaffer, and A. V. Balatsky
Dirac Magnon
AF Dirac Magnon
Dirac magnons are inherent to honeycomb lattice (geometric properties): Ferro or AF does not matter
In sharp contrast to cubic lattice
Ferromagnet Anti-ferromagnet
3-dim cubic lattice Since 1930
𝜔𝑘 ∝ 𝑘2 Non-relativistic
𝜔𝑘 ∝ 𝑘 Relativistic
2-dim honeycomb lattice [arXiv:1512.04902]
𝜔𝑘 ∝ 𝑘 Relativistic
𝜔𝑘 ∝ 𝑘 Relativistic
[arXiv:1512.04902] J. Fransson, A. M. Black-Schaffer, and A. V. Balatsky