There is an interesting paper
Quantum critical dynamics of a magnetic impurity in a semiconducting host
Nagamalleswararao Dasari, Swagata Acharya, A. Taraphder, Juana Moreno, Mark Jarrell, N. S. Vidhyadhiraja
The key physics of the Kondo model is the formation of a spin singlet state between the impurity spin and the spins of the electrons in the conduction band. We say, the impurity spin is “screened” by the spins in the conduction band.
The "screening" electrons involved span from the Fermi energy up to some higher energy.
The relevant energy scale is the Kondo temperature which depends in a non-analytic way on the density of states (DOS) at the Fermi energy, and is roughly the binding energy of the spin singlet.
As the DOS goes to zero the Kondo temperature goes to zero.
But, what if there is an energy gap at the Fermi energy, as in a semiconductor?
One might expect that the Kondo effect disappears and the local moment is no longer screened.
Specifically, is there a critical non-zero value of the energy gap below which the Kondo effect survives and one observes at Fermi liquid?
How about if the temperature is larger than the energy gap but less than the Kondo temperature?
Then perhaps the electrons that are thermally excited into the conduction band can screen the impurity spin.
The above fundamental questions are relevant to understanding magnetic semiconductors. They can be addressed by studying the gapped single impurity Anderson model. A number of numerical and analytical studies over the years have produced different answers to the above questions. The current paper gives definitive answers based on state-of-the art Quantum Monte Carlo calculations.
The phase diagram is shown below, with temperature versus the energy gap, delta.
Both are scaled by the Kondo temperature in the absence of the gap. LM denotes an unscreened local moment and GFL a Generalised Fermi Liquid.
It is not at all obvious (at least to me) that the universality of the delta=0 case has to extend to the non-zero delta case. But it does.
One sees that the critical value of the energy gap is zero.
Furthermore, above some non-zero temperature, of the order of a fraction of Kondo temperature and about one half of delta, a Generalised Fermi liquid forms where the local moment is completely screened.
The authors also show that the dynamic spin susceptibility associated the spin of impurity exhibits “quantum critical scaling” in the sense that it depends only on omega/T where T is the temperature and omega is the frequency.
Hopefully the paper will stimulate some experiments, either in quantum dots or in semiconductors, to observe this fascinating physics.