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Behind the Kondo effect: Why can a single impurity affect the properties of an entire metal?
In the world of metallic substances, a single impurity can possess unexpected powers. This phenomenon can be partly understood using the Anderson impurity model, a theoretical tool used to describe magnetic impurities embedded in metals. As the research deepened, scientists gradually understood how these impurities change the properties of the entire metal, thereby forming the Kondo effect.
Basic Concepts of Anderson Impurity Model
The Anderson impurity model was proposed by the famous physicist Philip Warren Anderson and is mainly concerned with describing magnetic impurities in metals. The model contains several key components, including the kinetic energy of the conduction electrons, a two-level term describing the energy levels of the impurities, and a mixing term coupling the conduction electron and impurity orbitals. In its simplest form, the Hamiltonian for this model can be written as:
H = Σk,σεkckσ†ckσ + Σσ εσdσ†dσ + U d↑†d↑< /sub>d↓†d↓ + Σk,σVk(dσ< /sub>†ckσ + ckσ†dσ)
In this model, c is the annihilation operator of conduction electrons, d is the annihilation operator of impurities, k is the wave vector of conduction electrons, and σ marks the spin. The parameters in the Hamiltonian include the Coulomb repulsion U of the impurity, and the coupling strength V.
Different working ranges
Depending on the relationship between the impurity energy level and the Fermi level, the Anderson model forms several different categories:
- Empty orbital region: When εd ≫ EF, there is no local magnetic moment.
- Intermediate regime: When εd ≈ EF, local magnetic moments may appear.
- Local magnetic moment category: When εd ≪ EF ≪ (εd + U) The generated magnetic moment will be Kondo screened at low temperature and transformed into non-magnetic many-body spin state.
Further studying heavy-fermion systems, scientists used the periodic Anderson model to describe the lattice structure of an impurity. This can help understand how f-orbital electrons interact with each other in heavy-Fermi systems under certain conditions. Its Hamiltonian form is:
H = Σk,σεkckσ†ckσ + Σj ,σεffjσ†fjσ + U Σjfj ↑ †fj↑fj↓ †fj↓ + Σj,k,σVjk(eikxjfjσ†ckσ + e< sup>−ikxjckσ†fjσ)
Here, xj is the position information of the impurity, and these complex interactions show that even at relatively long distances, f orbital electrons still have a profound influence on each other.
Development of the SU(4) Anderson model
In addition to the traditional Anderson model, there are many variants, such as the SU(4) Anderson model, which describes impurities with spin and orbital freedom and is particularly suitable for carbon nanotube quantum dot systems. . The Hamiltonian of the SU(4) model is as follows:H = Σk,σεkckσ†ckσ + Σi ,σεddiσ†diσ + Σi,σ,i′σ′(U/2)niσni′σ′ + Σi,k,σVk (diσ†ckσ + ckσ†diσ)
In this model, further coupling of spins and orbitals provides a deeper understanding of multi-electron systems.
ConclusionThe Kondo effect shows us that a single impurity in a metal can have a profound impact on the overall properties, thus giving rise to many subtle physical phenomena. Furthermore, through different models, we can gain a deeper understanding of these complex interactions and the theoretical basis behind them. So, how many more amazing discoveries like this will be waiting for us to explore in the future?
