ABSTRACT
In the previous chapter, we developed a mean-field criterion for local magnetic moment formation in a metal. As mean-field theory is valid typically at high temperatures, we anticipate that at low-temperatures, significant departures from this treatment occur. The questions we focus on in this chapter are: 1) how does the presence of local magnetic moments affect the low-temperature transport and magnetic properties of the host metal, and 2) what is the fate of local magnetic moments at low temperatures in a metal? These questions are of extreme experimental importance because it has been known since the early 1930s that the resistivity of a host metal, such as Cu with trace amounts of magnetic impurities, typically Fe, reaches a minimum and then increases as - In T as the temperature subsequently decreases. This behavior is illustrated in Fig. (7.1) for various Mo and Nb alloys [SCL1964]. A resistivity minimum and subsequent logarithmic-temperature dependence are in stark contrast to the resistivity of the pure metal that tends to zero monotonically as the temperature decreases. An additional surprise is that the — In T-dependence of the resistivity does not continue indefinitely to low temperature, but rather, below a characteristic temperature, the Kondo temperature, Tk , it phases out. Moreover, the spin properties of a magnetic impurity change fundamentally in the neighborhood of the Kondo temperature, as magnetic susceptibility measurements showed [H1969]. Well above the Kondo temperature, the magnetic susceptibility of the impurity spins obeys the Curie 1/T law for free magnetic moments. Below Tk , however, the susceptibility tends to a constant [H1969]. A constant susceptibility at T = 0 is characteristic of a singlet state polarized by a magnetic field (see Problem 7.1). Hence, in addition to being the temperature at which the — In T <target id="page_72" target-type="page">72</target>Resistance minima for Fe in a series of Mo-Nb alloys (from Sarachik et al., <italic>Phys. Rev</italic>. <bold>135</bold>, 1041 (1964)). https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429037245/cc3a6600-aaec-4350-ba37-6c8d1f17d018/content/fig7_7_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> behavior of the resistivity ceases, Tk , is the temperature at which the impurity and conduction electron spins begin to condense into singlet states. This condensation is complete at T = 0. The vanishing of the local moment below some characteristic temperature makes the Kondo problem fundamentally different from bulk ferromagnetism in which there is an onset of magnetism below some characteristic temperature, the Curie temperature. In the Kondo problem, just the opposite occurs; magnetism ceases at low temperatures.
