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Magnon
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A magnon is a quasiparticle, a collective excitation of the spin structure of an electron in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fixed amount of energy and lattice momentum, and are spin-1, indicating they obey boson behavior.

History

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Felix Bloch introduced the concept of a magnon in 1930[1] to explain the reduction of the spontaneous magnetization in a ferromagnet. At absolute zero temperature (0 K), a Heisenberg ferromagnet reaches the state of lowest energy (so-called ground state), in which all of the atomic spins (and hence magnetic moments) point in the same direction. As the temperature increases, more and more spins deviate randomly from the alignment, increasing the internal energy and reducing the net magnetization. Viewing the perfectly magnetized state at zero temperature as the vacuum state of the ferromagnet, shows the low-temperature state with a few misaligned spins as a gas of quasiparticles, in this case magnons. Each magnon reduces the total spin along the direction of magnetization by one unit of (the reduced Planck constant) and the magnetization by , where is the gyromagnetic ratio. This leads to Bloch's law for the temperature dependence of spontaneous magnetization:

where is the (material dependent) critical temperature, and is the magnitude of the spontaneous magnetization.

Theodore Holstein and Henry Primakoff,[2] and then Freeman Dyson further developed the quantitative theory of magnons, quantized spin waves.[3] Using the second quantization formalism they showed that magnons behave as weakly interacting quasiparticles obeying Bose–Einstein statistics for bosons.[4][5]

Bertram Brockhouse achieved direct experimental detection of magnons by inelastic neutron scattering in ferrite in 1957.[6] Magnons were later detected in ferromagnets, ferrimagnets, and antiferromagnets.

The fact that magnons obey Bose–Einstein statistics was confirmed by light-scattering experiments done during the 1960s through the 1980s. Classical theory predicts equal intensity of Stokes and anti-Stokes lines. However, the scattering showed that if the magnon energy is comparable to or smaller than the thermal energy, or , then the Stokes line becomes more intense, as follows from Bose–Einstein statistics. Bose–Einstein condensation of magnons was proven in an antiferromagnet at low temperatures by Nikuni et al.[7] and in a ferrimagnet by Demokritov et al. at room temperature.[8] In 2015 Uchida et al. reported the generation of spin currents by surface plasmon resonance.[9]

Paramagnons

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Paramagnons are magnons in magnetic materials which are in their high temperature, disordered (paramagnetic) phase. For low enough temperatures, the local atomic magnetic moments (spins) in ferromagnetic or anti-ferromagnetic compounds become ordered. Small oscillations of the moments around their natural direction propagate as waves (magnons). At temperatures higher than the critical temperature, long range order is lost, but spins align locally (in patches), allowing for spin waves to propagate for short distances. These waves are known as a paramagnon, and undergo diffusive (instead of ballistic or long range) transport.

The concept was proposed based on the spin fluctuations in transition metals, by Berk and Schrieffer[10] and Doniach and Engelsberg,[11] to explain additional repulsion between electrons in some metals, which reduces the critical temperature for superconductivity.

Properties

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Magnon behavior can be studied with a variety of scattering techniques. Magnons behave as a Bose gas with no chemical potential. Microwave pumping can be used to excite spin waves and create additional non-equilibrium magnons which thermalize into phonons. At a critical density, a condensate is formed, which appears as the emission of monochromatic microwaves. This microwave source can be tuned with an applied magnetic field.

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Magnon

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from Grokipedia
A magnon is a bosonic that represents the quantum of a , embodying the collective precessional excitation of spins in a magnetically ordered , such as a ferromagnet or antiferromagnet. These excitations arise from deviations in the aligned spin configuration and carry spin angular momentum without net charge transport, enabling low-dissipation propagation over micrometer to millimeter scales. In ferromagnets, magnons exhibit a quadratic at low energies, ωk≈Dk2\omega_k \approx D k^2, where DD is the spin stiffness and kk is the wavevector, allowing them to follow Bose-Einstein statistics and condense at low temperatures. The theoretical foundation of magnons traces back to 1930, when introduced spin waves in his seminal work to account for the temperature-induced reduction in ferromagnetic magnetization, predicting a T3/2T^{3/2} law for low-temperature behavior that aligns with experimental observations. In 1940, Theodore Holstein and Henry Primakoff advanced the framework by developing the Holstein-Primakoff transformation, which maps spin operators to bosonic , enabling the quantization of spin waves into discrete magnon excitations and facilitating perturbative treatments of interactions. This transformation, applied within the Heisenberg model of exchange interactions, revealed magnons as non-interacting quasiparticles in the linear approximation, though higher-order terms account for and . Magnons play a pivotal role in understanding thermal properties of magnets, such as the aforementioned Bloch T3/2T^{3/2} law for reduction due to thermal magnon excitations, and have been experimentally observed via neutron scattering, Brillouin light scattering, and ferromagnetic since the mid-20th century. In modern contexts, the field of leverages magnons for energy-efficient information processing, exploiting their ability to carry spin currents with minimal compared to electrons, as demonstrated in waveguide-based devices and hybrid magnon-photon systems. As of 2025, advances include magnon propagation in two-dimensional van der Waals magnets like CrI₃, standalone integrated magnonic devices enabling nanoscale , magnon in electron microscopes, and coherent control of magnon dynamics for quantum technologies. These developments position magnons as key elements in emerging spin-based computing paradigms, bridging fundamental with practical applications.

Introduction

Definition and Overview

A magnon is a that represents the quantum of a spin excitation, specifically a , in magnetically ordered materials such as ferromagnets. arise as coherent precessional motions of aligned magnetic moments within a crystal lattice, where neighboring spins deviate slightly from their equilibrium orientation and propagate through the material like a wave. This concept was foundational in early theories of , describing low-energy excitations above the ground state of the magnetic order. Analogous to phonons, which serve as the quanta of lattice vibrations in solids, magnons embody the quantized nature of these spin waves as discrete packets of energy and spin. Unlike phonons, however, magnons carry spin angular momentum, typically reducing the total by one unit of ℏ\hbar per excitation, making them key to understanding magnetic dynamics. This bosonic character allows magnons to follow Bose-Einstein statistics, enabling phenomena such as Bose-Einstein condensation under appropriate conditions. Magnons possess an integer spin of S=1S=1, classifying them as bosons that can occupy the same without restriction, in contrast to fermionic quasiparticles like electrons. Their emergence requires a magnetically ordered , where exchange interactions between spins stabilize the collective excitations, providing a framework for quantized descriptions within spin wave theory.

Importance in

Magnons play a crucial role in low-temperature , particularly in ferromagnetic insulators where they serve as the primary carriers below the . In these materials, the excitation of magnons contributes significantly to the specific heat, following a characteristic T3/2T^{3/2} dependence at low temperatures, which arises from the bosonic nature of these spin-wave quanta. This magnon contribution dominates over and electronic terms in insulating ferromagnets, providing a key mechanism for understanding thermal properties in magnetically ordered states. Near the Curie temperature TcT_c, magnons are intimately linked to magnetic phase transitions and , where fluctuations in the spin system lead to the breakdown of long-range order. As temperature approaches TcT_c, the magnon spectrum softens, and critical spin fluctuations—often described as overdamped magnon modes—drive the transition from ferromagnetic to paramagnetic phases, influencing exponents in critical scaling laws. These dynamics highlight magnons' role in mediating magnetic ordering and the emergence of collective behaviors at criticality. Beyond traditional magnetism, magnons exhibit interdisciplinary significance, bridging with , , and topological matter. In , magnons enable coherent coupling to spin qubits, facilitating transduction and entanglement in hybrid systems for scalable architectures. In certain theoretical models, magnon-mediated interactions can promote topological in systems like quantum wires coupled to helical magnets. Topologically, magnon excitations in lattices support protected edge modes and helical band structures, offering robust platforms for dissipationless spin transport. These connections underscore magnons' potential in advancing spintronic devices.

Historical Background

Early Theoretical Developments

The foundational theoretical understanding of magnons, or quantized spin waves, emerged from early efforts to explain ferromagnetism through quantum mechanical exchange interactions. In 1928, developed a model that attributed the alignment of spins in ferromagnets to quantum exchange effects arising from the and Coulomb interactions between electrons, providing the prerequisite framework for describing collective spin excitations. This Heisenberg model treated as resulting from symmetric exchange integrals that favor parallel spin alignments, laying the groundwork for subsequent analyses of spin dynamics in ordered magnetic systems. Building on this, introduced the concept of spin waves in his 1930 paper, describing them as classical, propagating deviations from the uniform magnetization in a ferromagnet. Bloch modeled the spin system as a lattice of coupled oscillators, where small deviations from perfect alignment propagate as waves due to nearest-neighbor exchange interactions. His analysis predicted the spin wave spectrum, characterized by a quadratic at low wavevectors, which demonstrated the thermal stability of by showing that low-energy excitations do not disrupt long-range order at finite temperatures. This work marked a pivotal shift toward viewing as a rather than isolated atomic moments. In 1935, and provided a phenomenological description of dynamics through their eponymous equation, which governs the precessional motion of the vector under effective fields including exchange and . The Landau-Lifshitz equation incorporates damping mechanisms and serves as a continuum limit for propagation, enabling predictions of resonance and relaxation in ferromagnetic media. This framework complemented Bloch's discrete lattice approach by offering a macroscopic tool to analyze wave-like behaviors in , influencing later quantizations of into magnons.

Experimental Milestones

The concept of magnons received indirect experimental support in through observations of curves in ferromagnets, which deviated from simple Curie-law behavior at low temperatures and aligned with predictions of thermal excitation of spin waves as proposed by Bloch. Post-World War II advancements in instrumentation enabled direct probes of magnetic excitations, with inelastic emerging as a pivotal technique for resolving magnon bands in ferromagnetic materials. A landmark achievement came in 1957 when Bertram N. Brockhouse reported the first direct observation of magnon dispersion using inelastic neutron on a of (Fe₃O₄), revealing quantized spin-wave excitations with energies matching theoretical expectations and confirming Bose-Einstein statistics for magnons. This work, conducted at , laid the foundation for mapping magnon spectra and earned Brockhouse the 1994 for developing neutron techniques. Subsequent studies extended these observations to metallic ferromagnets like iron, where dispersion relations were measured in the early , further validating spin-wave in simple cubic lattices. In the 1960s, electron spin (ESR) and ferromagnetic (FMR) techniques provided precise measurements of uniform magnon modes (k=0) in low- materials such as (YIG), enabling the study of long-wavelength excitations and their relaxation dynamics under fields. These experiments highlighted YIG's exceptionally narrow linewidths—on the order of 0.1–0.2 Oe at X-band frequencies—facilitating insights into intrinsic Gilbert and two-magnon processes that broaden lines. By the 1970s, uncovered magnon-phonon interactions through observations of hybridized modes and sidebands in antiferromagnets like FeBO₃, where level repulsion between one-magnon and phonon branches demonstrated strong coupling that altered scattering intensities and frequencies. These findings, building on earlier two-magnon Raman signals in MnF₂, established Raman as a complementary tool to neutron scattering for probing short-wavelength magnons and their lattice-mediated decay channels in insulating magnets.

Theoretical Framework

Spin Wave Theory

Spin wave theory establishes the classical and semi-classical foundations for describing coherent excitations of spins in magnetically ordered materials, such as ferromagnets, where deviations from perfect alignment propagate as wave-like disturbances. Building on Felix Bloch's seminal work, which introduced the concept of spin waves as quantized deviations in the ordered spin lattice, the theory treats these excitations as precessional motions of magnetic moments around their equilibrium direction. This approach is essential for understanding low-energy collective dynamics before advancing to full quantum treatments. The classical description originates from the Heisenberg Hamiltonian, which captures the dominant between localized spins:
H=−J∑⟨i,j⟩Si⋅Sj,H = -J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j,
where J>0J > 0 denotes the ferromagnetic exchange constant, Si\mathbf{S}_i are spin operators at sites ii and jj, and the sum runs over nearest-neighbor pairs. To derive the equation, the of the spins is governed by the Heisenberg equation of motion, dSidt=JℏSi×∑jSj\frac{d\mathbf{S}_i}{dt} = \frac{J}{\hbar} \mathbf{S}_i \times \sum_{j} \mathbf{S}_j, which, for small transverse deviations from the fully aligned (where spins point along the z-direction), linearizes into coupled equations for the transverse components SixS_i^x and SiyS_i^y. Assuming plane-wave solutions of the form Si⊥∝ei(k⋅ri−ωt)\mathbf{S}_i^\perp \propto e^{i(\mathbf{k} \cdot \mathbf{r}_i - \omega t)}, this yields the classical , describing harmonic oscillations with frequency ω\omega dependent on the wavevector k\mathbf{k}. In the long-wavelength approximation (ka≪1k a \ll 1, where aa is the ), the dispersion simplifies significantly, highlighting the roles of exchange and dipolar interactions in determining the . Exchange contributions dominate the quadratic term, arising from the nearest-neighbor that favors parallel alignment and resists spatial variations in spin orientation. Dipolar interactions, stemming from the magnetostatic fields of the spins, introduce long-range effects that modify the , particularly near k=0k = 0, by accounting for demagnetizing fields and shape ; in bulk samples, they often lead to a gap or elliptic , while in thin films, they enable surface modes. The resulting frequency takes the form
ω≈Dk2,\omega \approx D k^2,
where DD is the spin constant, conceptually representing the energy cost per unit curvature of the spin texture, with D∝JSa2D \propto J S a^2 from exchange ( SS being the spin magnitude) and corrections from dipolar terms scaling as μ0Ms2a3/S\mu_0 M_s^2 a^3 / S ( MsM_s the saturation ). To extend this classical picture toward quantization while maintaining approximate bosonic statistics for low-density excitations, the Holstein-Primakoff transformation maps the spin operators to creation and annihilation operators for bosons. Introduced in 1940, it expresses the spin components as Siz=S−ai†aiS_i^z = S - a_i^\dagger a_i, Si+≈2SaiS_i^+ \approx \sqrt{2S} a_i
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