Magnon

Magnon
classification
Boson quasiparticles
properties
electric charge	neutral
Dimensions	0 (theoretical)  kg
Spin	1
average lifespan	∞ (theoretically)

A ferromagnetic magnon in the semi-classical view as a spin wave

A magnon or magnon quasiparticle is a collective excitation state of a magnetic system with the properties of a boson quasiparticle. In solids, this state of excitation corresponds to the quantized form of a magnetic spin wave, analogous to the phonons as quantized sound waves .

In simple terms, it is a disturbance in the form of a deviation in the spin of individual particles, which propagates through the solid like a sound wave .

Basic statements

The prerequisite for the existence of magnons is the existence of a magnetic order , i.e. a coupling between the magnetic moments of the lattice atoms, which leads to preferred alignments of the moments to one another, e.g. B. parallel with ferro magnets or antiparallel with antiferromagnets .

The energy for wave-like excitations of these ordered moments is  quantized as with the elastic lattice vibrations (phonons). For the smallest possible excitation one chooses the term magnon , which is analogous to the phonon . In the usual semi-classical interpretation (see illustration), this magnon consists of a chain of spins rotating coherently in a certain way , since the energy is thereby reduced. In the ground state, for example, all spins point parallel upwards :

${\displaystyle |\psi {_{G}}\rangle =\left|{\mathord {\uparrow }},{\mathord {\uparrow }},...\right\rangle }$

In contrast, it shows in the quantum mechanical Magnonzustand that fits to this base state, at a single location - with a certain correlated probability that corresponds to the above semi-classical image - down :

${\displaystyle |\psi _{\vec {k}}\rangle ={\frac {1}{\sqrt {N}}}\sum _{j=1}^{N}\,e^{i{\vec {k}}\cdot {\vec {r}}_{j}}\,|...{\mathord {\uparrow }},{\mathord {\uparrow }},...,{\mathord {\uparrow }},{\stackrel {j}{\mathord {\downarrow }}},{\mathord {\uparrow }},{\mathord {\uparrow }},...\rangle }$

With

• ${\displaystyle {\vec {k}}\;\,{\hat {=}}\;\,(2\pi /\lambda )}$the wave vector of the magnon
  • λ is the wavelength
• N is the total number of particles
• e is Euler's number
• i the imaginary unit in the space of complex numbers
• ${\displaystyle {\vec {r}}_{j}}$is the position vector of the particle  j for which the spin is inverted.

This corresponds to the application of a magnon generation operator to the ground state: ${\displaystyle M_{k}^{+}}$

${\displaystyle |\psi _{\vec {k}}\rangle =M_{k}^{+}\,|\psi {_{G}}\rangle }$

with the atomic spin quantum number  S = 1/2.

The spin of the magnon, on the other hand, is always 1 - not only in the case of ferromagnets and atoms with half-integer spins - because the total spin of the system (in units of Planck's constant ) is reduced from to by the "carried magnon" . Because of this integer spin, the magnons are bosonic excitations. ${\displaystyle N\cdot S}$ ${\displaystyle N\cdot S-1}$

In 1999, Bose-Einstein condensation in a solid was observed on magnons for the first time , and in 2006 also at room temperature .

With ferromagnets

With ferromagnets , the simple model for small (large wavelengths) results in a quadratic dispersion relation (relationship between angular frequency and wave number): ${\displaystyle k}$

${\displaystyle \hbar \cdot \omega =4JS\left(1-\cos(ak)\right)\approx \underbrace {2\cdot J\cdot S\cdot a^{2}} _{D}\cdot k^{2}+{\mathcal {O}}(k^{4})=D\cdot k^{2}}$

via the exchange coupling of interacting spins ( amount , lattice constant ). ${\displaystyle J}$ ${\displaystyle S}$ ${\displaystyle a}$

The dependence on the wave number is therefore (here in the approximation smaller than  k ) quadratic as with “real” massive particles in the whole non- relativistic range (e.g. with neutrons ), although magnons like other bosonic quasiparticles have no mass.

In general, the dispersion relation is always direction-dependent ( anisotropic ). This can be easily observed through inelastic neutron scattering (the neutrons interact with the spins of the electrons and nuclei and thus measure the distribution of the magnetic moments of the electrons). Brockhouse first succeeded in detecting magnons in 1957. For D we get e.g. B. after Shirane u. a. a value of 281  meV  Å ² for iron. Magnon excitations from high-frequency alternating magnetic fields can also be observed in spin wave resonance experiments in thin layers .

Since one is dealing with a spontaneously broken symmetry with ferromagnets (the rotational symmetry is broken because a certain magnetization direction is distinguished), one can identify magnons as the goldstone particles assigned to the spin state , i.e. H. Excitations with low energy or (according to the dispersion relation) very long wavelength.

Magnons were first introduced as a theoretical concept by Felix Bloch . He derived a temperature dependence of the relative magnetization with an exponent 3/2 (Bloch's law), which was also confirmed experimentally. The magnetization is reduced by the heat-induced generation of magnons. ${\displaystyle T^{3/2}}$

Spin waves in ferromagnets were given further theoretical treatment by Theodore Holstein (1915–1985) and Henry Primakoff as well as Freeman Dyson in the 1940s and 1950s, who introduced boson transformations named after them.

With antiferromagnets

In antiferromagnetism , where magnetizations with opposite orientation exist on sublattices that penetrate each other, the magnon excitations have a completely different dispersion relation than with ferromagnets: here the energy does not depend on the square of the square , but - as with phonons - linearly on the wave number:

${\displaystyle \hbar \cdot \omega \propto k}$

This has u. a. concrete effects on the thermodynamics of the systems. So is z. B. in antiferromagnets the contribution of the magnons to the specific heat of a solid according to the Debye theory of the phonon contribution proportional to T ³ (T is the Kelvin temperature ) and can therefore only be separated from the contribution of the phonons by high magnetic fields.

Paramagnon

Paramagnons are magnons in the disordered (paramagnetic) phase of magnetic materials (ferromagnets, antiferromagnets) above their critical temperature. There only small areas are spin-ordered and allow the formation of magnons in these areas. The concept comes from NF Berk and JR Schrieffer and S. Doniach and S. Engelsberg, who used it to explain additional electron repulsion in some superconductors, which led to a lowering of the critical temperature.

See also

The magnons (more precisely: the underlying spin wave field) can also be described without direct reference to quantum mechanics using a classic non-linear integro differential equation system, see the vector Landau-Lifschitz equation . The actual magnons are described by quantum mechanics.

literature

• Charles Kittel Introduction to Solid State Physics. Oldenbourg Publishing House.
• J. Van Kranendonk, JH Van Vleck: Spin Waves . In: Reviews of Modern Physics . tape 30 , no. 1 , 1958, p. 1–23 , doi : 10.1103 / RevModPhys.30.1 .
• F. Keffer: In: S. Flügge (Hrsg.): Handbuch der Physik. Vol. 18, part 2, Springer, 1966.

Web links

• Physics News from Schewe / Stein on the Bose-Einstein condensation of magnons
• Phil Schewe, Ben Stein: First Bose-Einstein Condensate in a Solid. In: Physics News Update Number 746 # 2. September 21, 2005, archived from the original on December 31, 2005 . 
• Andreas Burkert: A magnon revolutionizes computer technology. In: microelectronics. Springer, August 22, 2014 . 

References and comments

1. ↑ The quasiparticles are almost all bosonic, for example phonons , magnons, polaritons , plasmons . But there are also fermionic quasiparticles, e.g. B. the polarons .
2. ↑ T. Nikuni, M. Oshikawa, A. Oosawa, H. Tanaka: Bose-Einstein Condensation of Dilute Magnons in TlCuCl ₃ . In: Physical Review Letters . tape 84 , no. 25 , 2000, pp. 5868-5871 , doi : 10.1103 / PhysRevLett.84.5868 .
3. ↑ T. Radu, H. Wilhelm, V. Yushankhai, D. Kovrizhin, R. Coldea, Z. Tylczynski, T. Lühmann, F. Steglich: Bose-Einstein Condensation of Magnons in Cs ₂ CuCl ₄ . In: Physical Review Letters . tape 95 , no. 12 , 2005, p. 127202 , doi : 10.1103 / PhysRevLett.95.127202 .
4. ↑ SO Demokritov, VE Demidov, O. Dzyapko, GA Melkov, AA Serga, B. Hillebrands, AN Slavin: Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping . In: Nature . tape 443 , no. 7110 , 2006, p. 430-433 , doi : 10.1038 / nature05117 .
5. ^ BN Brockhouse: Scattering of Neutrons by Spin Waves in Magnetite . In: Physical Review . tape 106 , no. 5 , 1957, pp. 859-864 , doi : 10.1103 / PhysRev.106.859 .
6. ↑ Kittel Introduction to Solid State Physics. 5th edition 1980, p. 553.
7. ↑ Kittel Excitation of Spin Waves in a Ferromagnet by a Uniform rf Field Physical Review , Physical Review, Vol. 110, 1958, pp. 1295-1297
8. ↑ F. Bloch: On the theory of ferromagnetism . In: Journal of Physics . tape 61 , no. 3–4 , 1930, pp. 206-219 , doi : 10.1007 / BF01339661 .
9. ↑ T. Holstein, H. Primakoff: Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet . In: Physical Review . tape 58 , no. 12 , 1940, p. 1098–1113 , doi : 10.1103 / PhysRev.58.1098 .
10. ↑ Freeman J. Dyson: General Theory of Spin-Wave Interactions . In: Physical Review . tape 102 , no. 5 , 1956, pp. 1217-1230 , doi : 10.1103 / PhysRev.102.1217 .
11. ↑ NF Berk, JR Schrieffer: Effect of Ferromagnetic Spin Correlations on Superconductivity, Physical Review Letters, Volume 17, 1966, pp. 433-435
12. ↑ S. Doniach, S. Engelsberg: Low-Temperature Properties of Nearly Ferromagnetic Fermi Liquids, Physical Review Letters, Volume 17, 1966, pp. 750-753
13. ↑ J. Miltat, G. Albuquerque, A. Thiaville: An Introduction to Micro Magnetics in the dynamic regime . In: Hillebrands B., Ounadjela K. (ed.): Topics in Applied Physics Vol. 83: Spin Dynamics in Confined Magnetic Structures I . Springer-Verlag, Berlin 2002, ISBN 978-3-540-41191-8 , pp. 1–34 , doi : 10.1007 / 3-540-40907-6_1 ( springer.com [PDF; accessed January 26, 2018]).