Kondo model

The Kondo model - also known as the sd model - is a mathematical model to describe the electrical resistance in metals with magnetic impurities - the so-called Kondo effect (the anomalous increase in resistance at very low temperatures).

In this simplified model, the electrons that generate electricity are modeled as free electrons in the conduction band (s-band). The magnetic impurities at position  i in the crystal lattice are assumed to be localized spins , which are coupled to the conduction band electrons via an (anti-) magnetic spin-spin interaction. The modeling of the magnetic impurities as localized spins is based on the assumption that the electrons are strongly localized in the d orbitals of the magnetic impurities. This sd interaction was first described in 1951 by Clarence Melvin Zener . Kasuya quantified this model in 1956 and established the associated Hamiltonian. In 1964 Jun Kondo treated this model using 3rd order perturbation theory and used it to calculate the electrical resistance. The calculated behavior of the electrical resistance qualitatively showed the experimentally found Kondo effect.

Mathematical description

The Kondo model can be described with the following Hamiltonian :

${\displaystyle {\begin{aligned}H_{\text{Kondo}}&=H_{0}+H_{J}\\H_{0}&=\sum _{{\vec {k}},\sigma }\epsilon ({\vec {k}})c_{{\vec {k}},\sigma }^{\dagger }c_{{\vec {k}},\sigma }\\H_{J}&=-{\frac {J}{N}}\sum _{n{\vec {k}}{\vec {k}}'}\exp(i({\vec {k}}-{\vec {k}}'){\vec {R}}_{n}){\vec {s}}_{\vec {k}}{\vec {S}}_{n}\end{aligned}}}$

This describes the conduction band electrons in the s-band with a dispersion relation . describes the interaction of the magnetic impurities - described via the localized spins in place with the conduction band electrons. The interaction is a pure spin-spin interaction with the spins of the conduction band electrons, which can be ferromagnetic or anti-ferromagnetic depending on the sign . ${\displaystyle H_{0}}$ ${\displaystyle \epsilon ({\vec {k}})}$ ${\displaystyle H_{J}}$ ${\displaystyle {\vec {S}}_{n}}$ ${\displaystyle {\vec {R}}_{n}}$ ${\displaystyle {\vec {s}}_{\vec {k}}}$ ${\displaystyle J}$

Results of perturbation theory

With anti-ferromagnetic coupling (negative J) in 3rd order perturbation theory, the Kondo model has a logarithmic term in the electrical resistance.

${\displaystyle {\begin{aligned}\rho _{\text{Kondo}}(T)\propto J/\epsilon _{f}\log T\end{aligned}}}$

Here is the Fermi energy . This logarithmic term thus leads to an increase in resistance at low temperatures and can thus explain the experimental data. However, this term diverges for what describes an unphysical behavior. This divergence is known as the Kondo problem. ${\displaystyle \epsilon _{f}}$ ${\displaystyle T=0}$

Individual evidence

1. ^ C. Zener: Interaction Between the $ d $ Shells in the Transition Metals . In: Phys. Rev. Band 81 , 1951, pp. 440-444 , doi : 10.1103 / PhysRev.81.440 .
2. ↑ T. Kasuya: A Theory of Metallic Ferro- and Antiferromagnetism on Zener's Model . In: Progress of Theoretical Physics . tape 16 , 1956, pp. 45-57 , doi : 10.1143 / PTP.16.45 .
3. ↑ J. Kondo: Resistance minimum in dilute magnetic alloys . In: Progress of Theoretical Physics . tape 32 , 1964, pp. 37-49 , doi : 10.1143 / PTP.32.37 .