View articles in the Special Topics Section:
Kondo Effect-40 Years after the Discovery
J. Phys. Soc. Jpn. 74 (2004) pp. 1-238
The Kondo Effect: from Resistance Minimum to Strong Electron Correlation
[January 14, 2005]
The resistivity of ordinary metals decreases with decreasing temperature, and tends to be constant at zero temperature. The temperature-dependent part of resistivity arises from scattering by lattice vibrations, and the constant part from the scattering by impurities. In the case of metals containing magnetic impurities, the behavior of resistivity is quite different. As temperature decreases, it first decreases and then increases, passing through a minimum. This phenomenon of resistance minimum was discovered in 1930's, and has been a mystery for a long time until the paper by Kondo was published. In his paper, starting from the s-d model in which an impurity spin and conduction electrons interact with each other by exchange interaction, Kondo showed that a logarithmic term exists in the t -matrix of scattering by the magnetic impurity in the second order of the exchange interaction. When the interaction is antiferromagnetic, the resisitivity due to this term increases as temperature decreases, and results in a resistance minimum. Kondo's theory was in an excellent agreement with experiments, as shown in Fig. 1.
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| Fig.1 Comparison of Kondo's theory with experimental data on resistivity of Au metals containing Fe impurities[1]. |
The logarithmic term diverges at zero temperature. When most divergent terms in each order are summed up, the series diverges at TK, TK being the Kondo temperature. This suggests that a strong correlation exists between the impurity spin and conduction electron spins at low temperatures. Yosida[2] showed that a singlet bound state is formed between them at zero temperature, where electrons behave as Fermi liquid[3]. It was finally proved using the exact solution of the s-d model[4].
Here, it is important to know how a magnetic impurity is formed. Anderson[5] proposed a simplified model of the magnetic impurity, in which a repulsive interaction acts between electrons localized at the impurity level with a finite width due to mixing with conduction electron levels. In this model, if the repulsive interaction is sufficiently strong compared with the level width, the impurity level cannot be occupied by two electrons with opposite spins, and then a localized spin is formed at the impurity site. The s-d model studied by Kondo corresponds to the limit of strong repulsion. On the other hand, in the noninteracting limit, the ground state is evidently a singlet as the level is occupied equally by up and down electrons. Thus, in both weak and strong interactions, the ground state has the same symmetry. This suggests that the noninteracting ground state of the Anderson model adiabatically continues to the singlet ground state of the s-d model as the repulsion increases. Actually, Yosida and Yamada[6] showed this explicitly by carrying out the perturbational calculation with the repulsive interaction. This is also proved using the exact solution of the Anderson model[7]. From a theoretical view point, the study of the Kondo effect based on the Anderson model is an ideal example in which the adiabatic continuation is exactly proved.
It should be noted that the existence of a sharp Fermi surface plays a crucial role in the Kondo effect. First, the logarithmic term Kondo found arises from it. The resistivity due to the impurity scattering increases with decreasing temperature, and tends to have a finite value, the unitarity limit. This indicates that the resonance scattering takes place on the Fermi surface. Correspondingly, a sharp peak in the density of states with a width of the order kBTK appears on the Fermi surface.
The study of the Kondo effect gives a reliable basis to the study of strong electron correlation in more general cases, e.g., the Hubbard model. The Hubbard model is a model simplified for the study of electron correlation. It is a nondegenerate tight-binding model of band electrons with a repulsive interaction between electrons at the same lattice site. If one lattice site is separated from the rest and the latter is replaced by its average, then the Hubbard model reduces to the Anderson model of an impurity whose solution is already known. Adopting the self-consistent condition as in the mean-field approximation, we obtain the approximate ground state of correlated electrons. The density of states thus obtained has a sharp peak on the Fermi surface, similar to the ground state of the s-d model. This is the dynamical mean-field theory, which is known to be exact in infinite dimensions[8].
In the papers collected in the Special Topics Section of J. Phys. Soc. Jpn. Vol.74 No.1 issue, various aspects of the Kondo effect, both theoretical and experimental, are discussed. This clearly shows that the physics of the Kondo effect is very fruitful and continuously applied even 40 years after the publication of Kondo's paper.
References
[1] J. Kondo: Prog. Theor. Phys. 32 (1964) 37.
[2] K. Yosida:Phys. Rev. B 147 (1966) 223.; Prog. Theor. Phys. 36 (1966) 875.
[3] P. Nozieres: J. Low Temp. Phys. 17 (1974) 31.
[4] N. Andrei et al.: Rev. Mod. Phys. 55 (1983) 331; A. M. Tsvelick and P. B. Wiegmann:Adv. Phys. 32 (1983) 453.
[5] P. W. Anderson: Phys. Rev. 124 (1961) 41.
[6] K. Yamada: Prog. Theor. Phys. 53 (1975) 970; K. Yosida and K. Yamada: Prog.Theor. Phys. 53 (1975) 1286.
[7] P. W. Wiegmann: Phys. Lett. A 80 (1980) 163; N. Kawakami and A. Okiji: Phys. Lett.A 86 (1981) 483.
[8] W. Metzner and D. Vollhardt: Phys. Rev. Lett. 62 (1989) 324.
Last updated 2005-1-14