| Stoner model | |
|---|---|
| Name | Stoner model |
| Introduced | 1930s–1940s |
| Inventor | Edmund C. Stoner |
| Discipline | Condensed matter physics |
| Subdiscipline | Magnetism |
| Keywords | Itinerant electron model, Ferromagnetism, Mean-field theory, Exchange interaction |
Stoner model
The Stoner model is a minimal theoretical framework in quantum mechanics and condensed matter physics that describes itinerant electron ferromagnetism in metals. It combines a single-particle description of electrons in a band with a local spin-dependent interaction to predict spontaneous spin polarization; the model established the concept of a criterion for ferromagnetic instability in terms of the density of states and an exchange parameter. The Stoner picture remains foundational for understanding magnetic metals, spintronic materials, and as a starting point for more sophisticated many-body treatments.
The Stoner model was introduced by Edmund C. Stoner and developed further in mid-20th-century studies of metallic magnetism. It addresses ferromagnetism arising from delocalized, or itinerant, electrons as opposed to localized-moment pictures such as the Heisenberg model and the Ising model. The model sits at the interface of single-particle electronic structure theories—notably the free electron model and band theory—and interaction-driven magnetism, invoking an effective on-site exchange to capture the tendency of electrons to align spins. It is commonly taught alongside Hund's rules and Pauli exclusion principle considerations and influenced developments in density functional theory (DFT) and many-body physics.
The minimal Hamiltonian of the Stoner model can be written as a tight-binding or continuum band term plus a contact interaction: H = \sum_{k\sigma} \epsilon_k c^\dagger_{k\sigma} c_{k\sigma} - I \sum_i n_{i\uparrow} n_{i\downarrow}, where \epsilon_k denotes the band dispersion from Bloch theorem / tight-binding model and I is the Stoner or exchange parameter representing intra-atomic exchange energy. In momentum space the interaction is typically treated as local (Hubbard-like) and spin-dependent. The model is closely related to the Hubbard model in the limit of weak correlations and large bandwidth; unlike the full Hubbard model, the Stoner model emphasizes mean-field exchange and often omits explicit charge fluctuations. Key quantities include the noninteracting density of states N(ε), the spin-dependent chemical potentials, and the magnetization m = n_\uparrow - n_\downarrow.
Applying a mean-field theory decoupling yields an effective single-particle picture with spin-split bands: \epsilon_{k\sigma}^\mathrm{eff} = \epsilon_k - I \langle n_{-\sigma}\rangle. Linearizing the self-consistency condition near zero magnetization produces the Stoner criterion for an instability toward ferromagnetism: I N(E_F) > 1, where N(E_F) is the density of states at the Fermi energy E_F of the noninteracting system. This criterion parallels concepts in BCS theory where an interaction and a density of states determine an instability threshold. The mean-field solution predicts the magnitude of magnetization and exchange splitting, and it establishes a Curie temperature estimate in the weak-coupling limit. Connections to Landau theory of phase transitions and the concept of a second-order magnetic phase transition are standard consequences.
The Stoner model provides a qualitative understanding of ferromagnetism in transition metals such as iron, cobalt, and nickel where d-band electrons are itinerant. It is used in interpreting results from angle-resolved photoemission spectroscopy (ARPES), magnetization measurements, and electronic structure calculations from density functional theory with local spin density approximation (LSDA). In spintronics research, Stoner-like exchange describes the spin polarization of conduction electrons in ferromagnetic electrodes and underpins models of giant magnetoresistance and spin transfer torque. The model also informs studies of magnetic instabilities in weak ferromagnets and alloys, and it is referenced in analyses of magnetic transitions under pressure or chemical substitution.
The simplicity of the Stoner model makes it analytically tractable but also limits its quantitative accuracy. It neglects strong electron correlation effects, dynamical spin fluctuations, and nonlocal exchange and correlation beyond on-site interaction. Extensions include embedding the Stoner concept into the Hubbard model, adding long-range Coulomb interactions, or combining with Dynamical mean-field theory (DMFT) to capture local quantum fluctuations. Spin-fluctuation theories (e.g., Moriya's self-consistent renormalization approach) and diagrammatic many-body techniques (e.g., random phase approximation, RPA) address deficiencies near quantum critical points. For narrow-band or strongly correlated materials, methods such as GW approximation, Bethe ansatz in 1D models, and cluster DMFT are necessary to go beyond the mean-field Stoner picture.
Experimental assessments of the Stoner model compare measured magnetic moments, exchange splittings, and Curie temperatures to predictions from band-structure calculations with an effective I parameter. Early successes include qualitative reproduction of ferromagnetism in 3d transition metal elements and trends across the periodic table. Discrepancies—such as reduced moments, temperature dependence of quasiparticles, and anomalies near quantum critical points—highlight the role of spin fluctuations and correlation effects. Modern probes relevant to testing and refining Stoner-based interpretations include inelastic neutron scattering for spin excitations, ARPES for band splitting, Mössbauer spectroscopy and nuclear magnetic resonance (NMR) for local fields, and transport experiments under pressure or doping. The Stoner framework remains a starting point for materials design in magnetocaloric materials, ferromagnetic semiconductors, and engineered heterostructures used in spintronics.
Category:Condensed matter physics Category:Magnetism (physics)