*2.3. Kondo-Fano E*ff*ects in Silicon Nanostructures*

Another interesting aspect of the Kondo effect is that it provides the opportunity, when the quantum channel of transport is coherent, to observe phenomena that have for a long time been confined to only the optical side of the physical experiments. In this context, silicon nano-systems like the ones discussed in the previous sections, have recently been able to demonstrate [10,11] that they can manifest the Fano effect, which consists in the interference of a discrete coherent channel with a continuum and which gives rise to characteristically asymmetric peaks in the response [10,11,36]. Another way to describe this effect is by mentioning that specially shaped asymmetrical peaks, in this

case current peaks, will appear every time with a path of rapid phase variations with almost constant phase interferences [10,11,36]. In a geometry like the one illustrated in Figure 2, the interference pathway can arise when more than one channel of conduction becomes available between the source and the drain terminals. This multi-channel configuration, in conjunction with the Aharonov-Bohm effect [37], allows the observation of the version of the effect that goes under the name of Fano-Kondo effect [10,11]. In the example discussed in References [10,11], the constant continuum is provided by a sequentially tunneling channel, while the coherent channel is provided by the Kondo effect due to correlated effect in the Coulomb Blockade regime as described in the previous section of this paper. Each of these two channels is linked to the transport via an atomic confinement potential located somewhere in the channel of the three-terminal device. A slave-boson mean-field approximation within the scattering matrix formalism was used to ensure that the data observed in Reference [10] was correctly interpreted within the framework of the Fano-Kondo effect [11].

### *2.4. Charge Pumping E*ff*ects in Silicon Nanostructures: Single Electron Pumps*

In 1983, D.J. Thouless published a seminal paper [38] predicting that a periodic perturbation of the confinement potential for electrons in a nanostructure would, in principle, allow the generation of topologically protected band levels. These levels could lead to energetic regions where the transport of electrons from the source to the drain could be precisely controlled, even when the voltage between the source and the drain is zero, which statement is an apparent violation of one of the fundamental principle of electronics, Ohm's law. By taking advantage of Coulomb Blockade phenomena's and of quantum tunneling effects [4,5], this concept was expected to enable the generation of ultra-accurate currents [12–15]. See Figure 6a for an illustration of a typical setup that can be used for the kind of measurements described in this section [12–15].

The quantification of the currents generated under the above-mentioned technique, which goes under the name of quantized pumping, is based on the formula ISource-Drain = *<sup>n</sup>*\**f*\**e*, see Figure 6b,c, where *e* is the elementary charge (1.6021766208 × <sup>10</sup>−19C), *n* is a positive integer indicating the number of transported electrons for each period or cycle, and *f* is the frequency of perturbation of the confinement potential, with *f* = 1/τ and τ being the period. This technique allows the implementation of high performant clocked single-electron sources, i.e., generating sufficiently high currents with sub-parts per million (sub-ppm) uncertainties [12–15]; thanks to shot noise being naturally suppressed in these experiments [39], while adverse temperature and flicker noise effects can be substantially suppressed [12–15]. The basic idea of these experiments aims at the generation of a current flowing through a quantum dot/single atom impurity without applying a voltage between the leads, but by specifically varying potential at one or more gates. Furthermore, as Figure 6a shows, the schematic of the most common devices that are used for charge pumping experiments are slightly more elaborated, if compared to the one discussed in Figure 2, see also the many device geometries described in Reference [5].

In Figure 6a, it is shown that at least two terminals are needed for an independent control of the two gates and, therefore, of an independent control of the in-to state and the out-from state tunnel barriers (i.e., Γin and Γout). This is different from the geometry schematically described in Figure 2 where only one terminal was used for the control of both these tunnel barriers. One of the most important aspects of this research is the fact that by operating the voltage gates in an opportune way, it is possible to obtain the appropriate sequential time-evolution of the voltages applied to the two gates, and therefore it is possible to generate an opportune time-evolution of the transparency of the tunnel barriers (i.e., <sup>Γ</sup>in,i and Γout,i) that allows the transport of exactly *n* electrons between the source and the drain each cycle [40]. The number of cycles per seconds (=*f*) will then determine the intensity of the current according to the law *I*Source-Drain = *<sup>n</sup>\*f\*e*. This description of the quantum pumping current clarifies why these experiments where the current is made by controlling electrons one-by-one are expected to generate precise/accurate currents. A sequence schematically describing the different sections of an ideal pumping cycle for a single atom pump [12,13] is set out under Figure 7, describing (a) the capture section of the cycle, (b) the isolation section, and (c) the emission section [13].

**Figure 6.** (**a**) Schematic of a typical measurement setup for a Single-Electron Pump [11–14]. (**b**) Characteristic *nef* response for *n* = 1 for a Single-Electron Pump [12]. (**c**) Characteristic *ef* response at different f (from a few MHz to 1000 MHz) for a Single-electron Pump [12].

Finally, if ideal quantum pumping transport is achieved, it is possible to obtain an ultra-fast and accurate circulation of an electrical charge (i.e., electrons) from the source to the drain [12–15,40]. The technical details on the manner in which these experiments can be performed is beyond the scope of my present review, see for example, the experiment descriptions in References [12,14,15]

I would like to conclude this review paper by describing in more detail the physics of the errors/non-ideal behaviors that can be observed in quantum pumps and I would like to discuss how these errors are, in particular, linked to silicon valley-orbital effects. Such errors can be seen as detrimental effects as they can cause the degradation of the currents measured in these charge pumping experiments. Interestingly, those experiments that were originally aimed at obtaining an accurate quantum definition of the ampere [12–15], more recently have opened up new directions in the fields of quantum information science [41–43] and classical signal processing [44].

### *2.5. Errors during the Operations of Single Electron Pumps*

To understand the mechanisms of operations of an ideal single electron pump, it is important to remind ourselves that the main result, which is required when these devices are in operation is to obtain the most accurate possible control of electrons during each one of the fast cycles that are used for the circulation of electrons from the source to the drain via the localized state. Hence, the mainstay of these experiments relies on the ability to obtain a synchronized control of all the tunneling rates; the ones that control the movement of electrons from the source to the ones of the levels of the localized state (i.e., Γin,i) and the ones related to the movement of electrons from the localized state to the drain (i.e., Γout,i) [12,13]. Of course, the different relaxation rates internal to the localized state will

also play a fundamental role in these experiments [12,13,15,25,45–52]. Here, it is important to mention that these experiments are mostly performed with source-drain bias at 0 V or in a region for which changes in the values of the source-drain bias are non-influent [12–15], i.e., the situation has given rise to the assumption that these pumping experiments represent a violation of an ideal definition of Ohm's law [38].

The central idea of these experiments is to obtain optimal dynamical control of the di fferent tunnel barriers that control the movement of electrons from source to drain. In turn, this is expected to lead to an optimization in the transport properties even when high frequencies, f, of operations are in use [12–15]. The conventional model describing the ideal operations of single-electron pumps, based on quantum dot (QD) confinement site, is often described as the Decay-Cascade model [46]. In this model, the pumping cycles can be schematically described with these three cyclical steps: (a) Capture from the source which for a QD is an easy occurrence because the dot is typically semi-open in the first section of the cycle [15,46]; (b) isolation into the localized state of *n* electrons; and (c) emission to the drain of *n* electrons, see also Figure 7 for a description of a similar cycle valid for a single atom pump [12,13]. If the ideal picture described above is followed during experiments, then the system will generate a current that is exactly equal to *<sup>n</sup>\*f\*e*.

However, I would mention that something could go wrong during each of the steps described above and the quantum pump could lose control of a certain number of electrons for each cycle. In turn, this can be the cause of degradation in the precision/accuracy of the current produced by the single-electron pump, and therefore this can lead to the generation of errors. As set out below, I have endeavored, based on the simplified schematic of Figure 7, to provide a simplified explanation to the most important mechanisms of errors that can be observed in these systems:


can lead to the delocalization of electrons between the ground-state and the excited states and as for QDs the rates that govern the tunneling between the excites state and the source/drain leads are fast, if compared to the ones between the ground state and the source/drain leads, hence, when electrons are delocalized, their probability of non-completion of the isolation step is much higher than normal [40,48]. Ultimately, this could lead to errors since it means that electrons will not be emitted to the drain and will not complete their cycle [48].

The detrimental mechanisms described above are much less likely to happen in SAPs as for these systems the shape of confinement potential is not as affected by thermal or non-adiabatic effects as much as it is in the case of QDs pumps.

**Figure 7.** Ideal pumping cycle for a single atom pump [12,13] showing a schematic description for the different steps; (**a**) the capture section of the cycle, (**b**) the isolation, and (**c**) the emission.

Consequently, even when they are operating at 4.2 K and at GHz frequencies of excitation, non-adiabatic effects are not as efficient in causing errors in SAPs, as opposed to QDs pumps [12,13]. Furthermore, for SAPs the relaxation rates (Γrelaxation, see also schematic in Figure 7a) controlling the relaxation of the electrons from excited states to the ground states are considerably faster [24] when compared to the equivalent ones observed in QD pumps [48]. Here, it is very important to emphasize that the fast relaxation rate effects observed naturally in SAPs are directly linked to the energy spectrum that the multi-valley physics imposes on these silicon systems [12,13,24]. Thus, for SAPs, the isolation step of the pumping cycle can always be reached efficiently. This also leads to a different way to operate these quantum pumps, based on the initial capture via excited state and sub-sequent fast relaxation to a well-isolated ground state [12,13]. As such, it is important to outline that the high performances observed in SAP's are linked to the indirect band gap properties characteristic of silicon materials. It is

also important to associate the non-adiabatic effects observed for *f* = 1/τ between 100 MHz [48] and a few GHz [45] to a recent set of results hinting to the ability of studying the ultra-fast coherent dynamics of electrons in highly reproducible silicon CMOS compatible devices [45].

(c) The alternative way to operate a SAP described above can be relatively error-free, unless these systems are excited to frequencies considerably higher than the GHz ones [12,13]. Consequently, the discussion above opens the way to the description of another kind of errors that could arise in QD or in single atom pumps [13] when electrons reach the confinement potential via an excited state and not the ground state. If the *f* = 1/τ approaches the values of Γrelaxation described in Figure 7, see also Reference [13]. In this situation, the electrons do not have sufficient time to relax to the ground state and the completion of the isolation step is compromised. The picture above can also be used to understand the causes of errors and of the degradation of the precision/accuracy of the measured currents in SAP's [13].

Other complicated versions of the kind of errors described under sections (b) and (c) above have been observed in other systems [50–52] and are often linked to the fact that more than one confinement site is playing a role in the control of electrons [50–52]. It is important to remind that multi-valley physics effects play an important role in the dynamical evolution of all these errors, see for example Reference [13].

(d) Lastly, I would like to briefly discuss another possible mechanism of error that can cause the degradation of the current and which has recently been observed in a silicon QD system [15]. For a system where a QD pump is operating at ultra-fast frequencies of excitations (up to 3.55 GHz), it has been shown that the ideal behavior of the pump can sometimes be affected by errors that appear when an impurity-trap state can compete with the main QD in the capture and in the emission of the electrons [15]. Note that the eventual presence of impurity-trap states in the gate stack of silicon devices is a well-known fact [1,4,45]. This novel frequency dependent mechanism [15], has not been completely explained, and it is a reminder that for silicon CMOS compatible technology, although extremely controlled and reliable [4,5], it is still possible to observe some unexpected behaviors. It is however comforting to note that the hybrid dot-impurity systems, such as the one discussed in this Refs. [15,45], have recently been able to provide record high performances in term of frequency and accuracy of operations [15,50–53], but have also opened up the way to the use of the quantum pumping technology for novel quantum information schemes [45].
