More about These Kondo Effects

In this section, by observing the schematic of Figure 4, I will describe the main features that can be associated with the different versions of these Kondo effects.

In its most conventional manifestation, the Kondo regime of transport is linked to the availability of the electronic spin degree of freedom only [32]. In this situation, this effect is suppressed as soon as a sufficiently high magnetic field is applied to the system. This is schematically illustrated in the diagram of Figure 4a, which shows that when a dual spin degenerate level is available and if there is no external magnetic field applied, then quantum fluctuation (schematically described in Figure 4 with the dashed

loops when observable in the current signature) can be effective in allowing transport in the Coulomb Blockade forbidden regions. Figure 4b also illustrates that, when a sufficient external magnetic field is in place and if the two spin levels are sufficiently separated energetically due to Zeeman splitting, fluctuation effects cannot be sufficiently efficient in generating an observable current [8–11,32,33], i.e., the Kondo effect is indeed suppressed. This effect is also observable in the experiments [32].

**Figure 3.** Example of Coulomb Blockade data similar to the ones in [7,8] and taken at 290 mK in a three-terminal device as the one schematically described in Figure 2. As an example, according to the orthodox Coulomb blockade theory, transport should arise only in the region delimited by the triangles contained within the pink dashed lines. Any transport signature outside these regions is linked to higher order effects [4,8,9]. An example of the latter is the signal present in the low bias, i.e., |VSource-Drain| < 5 mV, region between the one-electron (i.e., D0) and the two-electron (D−) charge states [4]. This signal, also outlined with the rectangular transparent area, is linked to the observation of spin and orbital Kondo fluctuations. The low bias region after the two-electron D− is most likely linked to the occurrence of the Kondo based on fluctuation of integer degree of freedom [34].

The simple picture introduced in the section above and in Figure 4a,b can become more complicated for systems for which the valley orbit pseudo-spin degree of freedom is also available for Kondo fluctuations, in the same manner as silicon systems, as illustrated in Figure 1 [8–11].

For the latter, a very unusual variety of Kondo, i.e., pure orbital Kondo, can also be observed [8–10,19,33], because valley-orbital pseudo spin is only lightly affected by magnetic field splitting. The occurrence of pure orbital pseudo-spin [19], i.e., not involving fluctuations of the spin of the electrons, Kondo effect in silicon nanostructures, is discussed here below:

(a) For conventional semiconductors, the Kondo effect has only been observed in relation to interactions between the spin of electrons confined within the localized state and the ones of the surrounding free electrons at sufficiently low temperatures (T's), i.e., for Temperature < Kondo temperature (TKondo), see Reference [32]. This situation is illustrated in Figure 4a, and as shown in Figure 4b, in this situation, the spin-Kondo effect is suppressed when a sufficiently high magnetic field is applied to the system because the Zeeman splitting between the spin up and the spin down of the electrons makes energetically impossible for spin fluctuations to generate virtual states that would open the Kondo transport channel [32]. This situation goes also under the name of conventional SU(2) Spin Kondo effect.

(b) In materials were the pseudo-spin degree of freedom is also available, however, as in the area in Figure 3 outlined by the rectangular shaded shape and as illustrated in Figure 4c, the Kondo effect is somehow different to the one shown in Figure 4a and as described in the above sections. A different situation from the one above has recently been observed and is evident both from the experimental and from the theoretical points of view in silicon CMOS three terminal devices [8,9]. The extension of the Kondo effect to valley-orbital degree of freedom is clearly illustrated in Figure 4c by introducing different colors (black and red) for the two-different valley-orbital levels involved in the effect, i.e., the two lowest states, as shown as degenerate in Figure 4c,d. Consequently, the Kondo effect observed in silicon nanostructures is a more sophisticated phenomenon that goes under the name of SU(4) Kondo effect [4,8,9].

**Figure 4.** For conventional Quantum Dot Systems where only the fluctuations of spin degree of freedom are available [32], the Kondo effect is suppressed for sufficiently high magnetic fields (BC) because of the Zeeman splitting makes energetically impossible for Kondo fluctuations to arise. Opposite to this, when observed in an opportunely tuned silicon system, the Kondo Effect can arise as the combined action of the fluctuations of the spin and of the pseudo-spin (Valley-Orbit) degrees of freedom. As the pseudo-spin is typically only lightly affected by the magnetic field, a pure Orbital version [7,8,24,32] of the Kondo effect survive even for B > BC.

What makes these latter results even more interesting is the fact that a previously unobserved Kondo ground state symmetry crossover can be studied in these systems [8], because, as from one side it is possible to saturate the spin degree of freedom by mean of Zeeman splitting, as it is shown in Figure 4b in conventional Kondo systems, and from another side it is possible to be in a situation where the pseudo-spin fluctuations are still in place even if under the effect of a sufficiently high magnetic field, B > Bc, with Bc being a certain critical field [8]. As shown in Figure 4d, in this situation, it is possible to saturate only the spin degree of freedom, but also to observe a Kondo effect originating only from the valley-orbital quantum fluctuations, the so-called "pure" SU(2) orbital Kondo effect [8,9,19]. These results imply pure quantum screening of the orbital degree of freedom [8].

Due to the difficulty of accessing these regimes, the observation of these novel effects represents not only an important milestone for quantum many-body physics, but has also opened-up new pathways for silicon quantum electronics such as, for example, valley-orbital quantum bits [4,20]. Of course, these many-body effects cannot be directly used for the implementation of novel quantum information schemes, but their observation is very important, because it has permitted the study and the characterization, in a coherent fashion, of quantum states based on these pseudo-spin degrees of freedom that could, in the future, become the base for novel quantum schemes [20]. What makes the transition between the different versions of the Kondo effects, as illustrated in Figure 4c,d, ever more interesting is that this transition represents a universal phase transition equivalent to the ones observed in other systems having the same symmetry in the quantum degrees of freedom, e.g., some nuclear systems [35]. As shown under Figure 5, in this latter situation, a survival of a TK - 0 and a constant evolution of this TK - 0 are observed even for B > Bc. While for conventional SU(2) Kondo for B > Bc TK is = 0.

**Figure 5.** Universal law of the smooth transition between two different versions of the Kondo Effect (because observable in systems very different one from of each other [8,35]) under the effect of an external magnetic field (i.e., from an SU(4) Kondo to an SU(2) Kondo effect). Even if this data does present some scattering, nevertheless, an initial linear behavior of the order parameter (Kondo Temperature, TC) can be observed between 0 Tesla and 2 Tesla. This is most likely followed by a constant value of TC for any B > 2 Tesla.

In conclusion, as also shown in Figure 5, by looking at the evolution of the order parameter of systems that have both spin and pseudo spin degrees of freedom (i.e., TK in the case of our systems), it is possible to observe a unique universal law for the smooth transition between two different versions of the Kondo effect under the influence of an external magnetic field. These TK's can be extracted by fitting the curves that describe how the current signature evolves at different temperatures [8]. Even if the data in Figure 5 does present some scattering, nevertheless, the experiment opens up an unique window to the characterization of universal physical behaviors that are expected to be observable in different systems [8,9,35].
