Tru Physics

Kondo Effect

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Introduction

The Kondo effect, named after the Japanese physicist Jun Kondo who first explained it in 1964, is a peculiar phenomenon in solid-state physics where the electrical resistance of a metal, containing a small number of magnetic impurities, increases as the temperature decreases. This behavior contradicts the typical understanding that the resistance of metals decreases as temperature decreases.

Magnetic Impurities and Spin Exchange

Magnetic impurities in a metal have a significant role in the Kondo effect. These impurities, with unpaired electrons, introduce local magnetic moments in the lattice of non-magnetic metals. The spin of conduction electrons in the metal interacts with these magnetic moments through a process called spin exchange interaction, leading to the Kondo effect.

Spin Exchange Interaction

In a metal with magnetic impurities, conduction electrons passing near an impurity can have their spin flipped due to a quantum mechanical interaction. This spin-flip scatter increases the electrical resistance, with the degree of scattering dependent on temperature. The mechanism is mathematically expressed by the Kondo Hamiltonian:

H = \displaystyle\sum_{k,\sigma}\epsilon_{k}c_{k,\sigma}^\dagger c_{k,\sigma} - J\vec{S}\cdot\vec{s}(0)

where c_{k,\sigma} and c_{k,\sigma}^\dagger are the annihilation and creation operators of a conduction electron with momentum k and spin \sigma, \epsilon_{k} is the electron energy, J is the exchange coupling constant, \vec{S} is the impurity spin operator, and \vec{s}(0) is the spin density of conduction electrons at the impurity site.

The Kondo Temperature

The Kondo temperature (T_K) is a characteristic temperature below which the Kondo effect becomes significant. It is defined by the coupling strength between the impurity spin and the spins of the conduction electrons and can be approximated by:

T_K = De^{\left(\dfrac{-1}{\rho J}\right)

where D is the conduction band half-width, \rho is the density of states at the Fermi level, and J is the exchange coupling constant.

Kondo Resonance and the Kondo Cloud

At temperatures below T_K, a unique phenomenon known as Kondo resonance occurs, leading to an anomalous increase in electrical resistivity. This is a result of the formation of a many-body quantum state, known as the Kondo singlet or Kondo cloud, around the magnetic impurity. The Kondo cloud effectively screens the magnetic impurity’s spin, resulting in a sharp resonance peak at the Fermi energy.

Conclusion

The Kondo effect is a fascinating and counterintuitive phenomenon in condensed matter physics, demonstrating the richness of quantum many-body systems. Its discovery has greatly influenced the development of many-body quantum theory and provided crucial insights into the physics of impurities in metals. Ongoing research seeks to explore this effect further in the contexts of quantum dots, heavy fermion materials, and high-temperature superconductors.

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