**2. Introduction to New Results**

This review paper discusses how silicon CMOS compatible devices have recently revolutionized not only the fields of quantum information and quantum communications, e.g., see Ref. [4,5], but also those related to mesoscopic physics because of their demonstrated ability to compete with exotic, but not as ye<sup>t</sup> commercially available materials as required for larger-scale applications, such as graphene [6] or carbon nanotubes [7]. Bearing in mind that silicon CMOS compatible devices remain the core platform for most of the traditional information processing infrastructures, this new development represents an impressive achievement [4,5].

To provide further details, this review paper aims at discussing how silicon CMOS compatible ultra-scaled devices can be used for the study of very elusive problems, for example the orbital Kondo effect [8,9], the Kondo-Fano e ffects [10,11] as well as the mechanism of errors in single electron pump (SEP) devices [12–15].

One of the interesting aspects is that these silicon-based devices possess some peculiar and unique properties, if compared to the ones present in other semiconductor materials, due to the fact that, for silicon-based materials, the electron band gap exists in an indirect form [4,5]. These properties can be summarized by using the schematic picture of Figure 1 and by mentioning that electrons in silicon possess an extra degree of freedom, known as the "valley-orbital pseudo spin degree of freedom" [4,8–11,16,17], which occurrence is related to the fact that, for bulk silicon materials, the energies of electron forms in the conduction band (CB) are not minimized when the crystal momentum **k** is equal to 0 and are 6-fold degenerate at the minimum point [4,8–11,16,17]. The use of the term "pseudo-spin" in conjunction with the terms "valley" and "orbital" is somewhat controversial. Nevertheless, it is often used in many recent publications, see for example References [4,8–11,16,17] However, in this document, I have repetitively made use of this term in the context of silicon valley-orbital degrees of freedom.

Consequently, the pseudo-spin can act as a good quantum number [18,19], and as Figure 1 schematically shows, the level degeneracy observed at the minimum of the CB can be partially or completely lifted by means of confinement e ffects see References [1–4] and references therein; hence, this degree of freedom can be utilized as a platform for novel quantum logic operations, see References [4,5] and references therein.

As an example, qubits based on the valley-orbital degree of freedom can be better isolated against the deleterious e ffects of charge noise as opposed to the ones based on more conventional degrees of freedom such as spin or charge [20]. Even though it is true that devices utilizing valley-orbit states are limited by the fact that deterministic control of the valley splitting is far from being routinely achieved [4,20]. In this review paper, I will show that, although the indirect bandgap properties of silicon were for a long time considered as a negative attribute of silicon materials, especially from the optoelectronic point of view, more recently these properties have demonstrated to be silicon material strengths, especially in view of the possible uses of silicon devices for quantum applications [4,5]. This is particularly true because of the numerous significant advancements made in the technologies used in the fabrication of silicon CMOS compatible devices [5]. In view of this, it is now possible to obtain extremely precise control of the quantum properties of electrons confined in silicon CMOS compatible structures even during fabrication [4,5] and it is also possible to engineer devices precise to a single atom level [21–24]. These new enhanced fabrication capabilities have translated into an improved ability for the control of all the quantum degrees of freedom (for example spin, charge, pseudo-spin) of electrons [8–17,20–24] and holes [25] in silicon nanostructures [4,5]. Finally, although there are still a few open questions in this area, for example see Reference [4], the achievements described above have translated into improved control of the electronic signature that can be observed in these devices. This, in turn, explains why these systems have also been used recently as platforms for the observation of some of the most remarkable e ffects in physics [4,8–15], in the same manner in which they were previously observed in other materials such as carbon nanotubes and graphene [6,7].

**Figure 1.** Schematics of the bottom of the conduction band (CB) for a few typical Silicon systems. The effects of confinement, of electrical effects and of structural atomic effects to the CB structure are included in (**a**) for two-dimensional Quantum Well and on (**b**) for an isolated dopant-atom impurity such as arsenic (As) or phosphorous (P), see also Reference [4]. Two-fold spin degeneracies are not included in this illustration [4].

### *2.1. Special Properties of the Electrons in Silicon CMOS Compatible Devices*

As already mentioned above and to be more specific regarding the characteristics of these silicon-based systems, it is important to note the fact that the energetic structure at the minimum of the conduction band of silicon devices fabricated on basic structures of quantum dots and single atom nanostructures can be quite different when compared to the energetic structures observed in bulk materials [1–4], see the schematic picture in Figure 1.

This observation justifies the proven capability of silicon to host nano-devices, e.g., see schematic in Figure 2, which allow a good control of the spin degree of freedom of electrons, due to its weak spin-orbit coupling and to the existence of isotopes of silicon with zero nuclear spin [4]. Hence, by using silicon devices, it is possible to engineer a well-controlled quantum environment that can act as the ground for the observation of several exciting mesoscopic physics effects. These phenomena will be the focus of this review paper.

**Figure 2.** Schematic of a three-terminal device with one or more than one gate controlling the tunneling barrier from the source to the state, <sup>Γ</sup>in,i, and the tunneling barrier from state to the drain, <sup>Γ</sup>out,i. The leads (source/drain) represents an infinite reservoir of electrons with the all the possible spin (si) and valley-orbital (pi) polarizations. The transport in the device can be controlled by applying a voltage to the Vgate terminal that controls the position of the quantum states in the confinement potential respectively to the Fermi level in the source and drain leads. In this configuration, the Vgate terminal can also control the transparency of the tunneling barriers <sup>Γ</sup>in,i and Γout,i. The VSource-Drain voltage at the drain terminal can control the polarity and the intensity of the current of electrons, while the source terminal is connected to a pico-ammeter. As the figure shows, all the elements of this circuit are connected to the same reference grounding.

Several recent review papers and books, i.e., see References [4,5,26,27] and references therein, have described in detail many of the different approaches that can be used to fabricate these silicon devices. Consequently, my current review will simply refer to the relevant publication and, in instances where it is necessary to elaborate on a particular point, I will provide additional information covering fabrication techniques, without repeating what has already been reported in the relevant publications such as References [4,5,26,27].

In order to understand the peculiar behavior observable with silicon nanostructures [4,5,26,27], it is important to initially describe in detail the structure of the minima of the conduction band introduced above, see Ref. [1–4] and Figure 1.

As silicon has an indirect band gap [1,2] and because of its lattice symmetry, its conduction band minima has six degenerate minima (or valleys) at the point k = 0.85 k0 of energy vs. wave vector, **k**, diagram in the reciprocal space, as also schematically shown with the black lines in Figure 1, with k0 being the wave vector that defines the size of the unit cell in the reciprocal lattice space of the material [1,2]. This peculiar band structure, unlike other materials that have a direct band gap, e.g., gallium arsenide (GaAs), produces a situation in that electrons in silicon have an extra degree of freedom, which can be used for their quantum dynamical control; i.e., the valley-orbital pseudo-spin degree of freedom introduced above. As such electrons in silicon are said to be affected by multi-valley physics [4].

Furthermore, because, in these structures, most of the low temperature transport of electrons is located around the minimum of the conduction band, the multi-valley physical properties described in Figure 1 play a critical role for most of the mesoscopic low-temperature effects, as described in this review paper. Moreover, I have no desire to cover in the extensive amount of materials represented by the more than fifty years of research in the fields of silicon microelectronics [1,3] and silicon nano-electronics [4]. However, I would like to discuss some interesting and more recent progress in the field of silicon nano-electronics [4]. Moreover, another interesting aspect is that, as shown in Figure 1, both for donor impurities in silicon, such as arsenic or phosphorous, for two-dimensional (quantum wells) and for zero-dimensional hetero-structures (i.e., quantum dots), due to the severity of effects such as confinement, electric fields with strong gradients, lattice imperfections, and atomic-scale details at the interface between different sections of a device, most of the energy levels degeneracy at the minimum of the conduction bands can be lifted. For a Quantum Well (QW), the 6-fold valley degeneracy is broken by the large in-plane tensile strain to a 2-fold degenerate Γ levels that are below 4-fold degenerate Δ levels [4]. These degeneracies can in turn be broken by all other atomic effects that go under the name of valley-splitting see Figure 1a). For isolated dopant-atoms in the silicon lattice, i.e., single atom transistors [8–13,17,21–24], in the most ideal situation, the final configuration gives a 1-fold degenerate 1s state (A1), below a 3-fold degenerate 1s and another 2-fold 1s state, as shown in Figure 1b), while other intermediate situations are possible [8,9]. As an example, the lower energetic valley-orbital states are fundamental for the observation of the Kondo and the Kondo-Fano effects described in the section below. These properties are also a fundamental ingredient for the successful operation at high frequency (≥GHz) of silicon based quantum electrons pumps [12–15].

The lifting of the valley-degeneracy of the electrons in quantum dot or quantum well devices in silicon can have multiple consequences; first it can allow the observation of novel quantum effects [8–15]. Furthermore, the study of this novel degree of freedom has demonstrated that this valley (pseudo-spin) degree of freedom can also act as a good quantum number [18,28] for electrons, and as such, it could be used for their coherent control in the implementation of quantum gates [4,20,28].

It is also important to clarify that, when electrons are confined in isolated dopant atoms or QD nanostructures an orbital order for the energy levels of the states also appears [4]. This orbital order in the hierarchy of the states can co-exist with the hierarchy of the distribution imposed by the physics of valleys [4]. Sometimes valley and orbital effects can lead to very distinguishable consequences in the structure of the energy levels [4], however, because these effects can typically be influenced in many ways by the microscopic structures of these devices, the most common way for valley/orbital effects to

manifest themselves in silicon nano-structures is in a mixed configuration [4]. This explains why, for most silicon nanostructures, the term "valley-orbit degree of freedom" is often in use as opposed to the terms "valley degree of freedom" or "orbital degree of freedom" [20,28]. It is also interesting to briefly mention that the signature of these complex valley-orbit effects imposes selection rules that lead to dramatic alterations of the observed electronic effects [28].
