How to determine the Kondo energy scale for an experimental result


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14 days ago by
In experiments on Anderson/Kondo materials, there is always a need to determine/assign a Kondo scale/temperature to the results.
Often, the system under consideration is very complex and a theoretical approach is either not available or insufficient. In such cases experimentalists have devised several ways to determine a Kondo energy scale (or temperature), and it is very important to understand such methods to avoid misunderstanding or nonsensical claims. Such ad-hoc methods cannot be used as an “explanation” but just as aids for a better understanding and intuition about the system — they cannot be used as replacements for proper theoretical justifications and analytical calculations.

One of the methods used in such experiments is to compare the observed, presumably, Kondo resonance with the resonance of a single-impurity Kondo system at the same ambient temperature as that of the experiment. Namely, the question is to find the best fitting curve \( f(T; T_K) \) to the observed data, where \( f \) is an approximate form of the Kondo resonance for a 1-impurity Kondo system with \( T_K \) as its energy scale; \( f \) depends on the temperature (and probably, other factors like voltage) (see Fig. 1). Note the important fact that \( T_K \) is an intrinsic property of the Hamiltonian; a dynamic energy scale which depends on the parameters of the Hamiltonian, not temperature. It is finite even at vanishing temperatures, as an NRG zero-temperature calculation can show explicitly (see e.g., Hewson, “The Kondo problem to heavy fermions”).



For instance, if we have an experimental data as in Fig. 2 at \( T \) = 10 K, the closest 1-impurity resonance at the same temperature, is a curve \( f \) with, e.g., \( T_K \) = 4.56 K.


In plain words, this implies that the Kondo mechanism in the system under consideration is effectively similar to that of a 1-impurity Kondo system with \( T_K \) = 4.56 K.
This is quite similar to what one does for determining the strength of magnetic moments in a material: We say a material has some magnetic moment of magnitude 1.78, which means, in effect, that if we compare/fit the magnetization of that material (at temperature \( T \)) with that of a simple Curie-Weiss model (at the same temperature), we obtain a value 1.78 for the magnitude of the magnetic moment. We do the same for the Kondo singlets to acquire an intuitive understanding.

Notice that if we measure the experimental resonance at several temperatures, we might obtain a different fit for each temperature, and hence, a different effective Kondo scale for the system as a function of temperature. That, indeed, does not mean that “Kondo temperature depends on the temperature”, which is sheer naïveté, but implies the effective strength of the local Kondo coupling is changing as temperature (or any other control parameter) is varied.

So, regarding the discussion above, I believe that statements like “measuring the Kondo weight vis-à-vis the Kondo temperature” or “measuring both simultaneously” barely make any sense, if at all.

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