# Unusual Quantum Transport Mechanisms in Silicon Nano-Devices

## Abstract

**:**

## 1. Introduction

## 2. Introduction to New Results

**k**is equal to 0 and are 6-fold degenerate at the minimum point [4,8,9,10,11,16,17]. 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 [4,8,9,10,11,16,17] However, in this document, I have repetitively made use of this term in the context of silicon valley-orbital degrees of freedom.

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

_{0}of energy vs. wave vector,

**k**, diagram in the reciprocal space, as also schematically shown with the black lines in Figure 1, with k

_{0}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].

#### 2.2. Kondo Effects in Silicon Nanostructures

_{Source-Drain}. The third terminal is typically used to apply a voltage to the V

_{Gate}and in turn, to control the transition of the device between the ON state and the OFF state at room temperature [1,29,30] and the position of the quantum level that is present in this system at low temperatures when Coulomb effects are observable [4,29,30]. For geometries, like the one shown below in Figure 2, the V

_{Gate}can in principle also control the transparency of the tunneling barriers Γ

_{in,i}and Γ

_{out,i}. While only more complicated structures, for example see the double gate structure, as discussed later on in this review paper, may offer the possibility for an independent control of the position of the quantum states and of the transparency of Γ

_{in,i}and of Γ

_{out,i}, even a simple structure like the one described in Figure 2, it is possible to observe many-body effects at sufficient low-temperatures [8,9,10,11,12]. In this review paper, I will often use this geometry as a platform for the description of the interesting effects that can be observed in silicon nano-devices, since the said platform has already demonstrated the capability to offer access to the quantum systems, as illustrated in Figure 1.

#### More about These Kondo Effects

- (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 (T
_{Kondo}), 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].

_{c}, with B

_{c}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].

_{K}≠ 0 and a constant evolution of this T

_{K}≠ 0 are observed even for B > B

_{c}. While for conventional SU(2) Kondo for B > B

_{c}T

_{K}is = 0.

_{K}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 T

_{K}’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].

#### 2.3. Kondo-Fano Effects in Silicon Nanostructures

#### 2.4. Charge Pumping Effects in Silicon Nanostructures: Single Electron Pumps

_{Source-Drain}= n*f*e, see Figure 6b,c, where e is the elementary charge (1.6021766208 × 10

^{−19}C), 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 [12,13,14,15]; thanks to shot noise being naturally suppressed in these experiments [39], while adverse temperature and flicker noise effects can be substantially suppressed [12,13,14,15]. 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}and Γ

_{out}). 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., Γ

_{in,i}and Γ

_{out,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}= n*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].

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

_{in,i}) and the ones related to the movement of electrons from the localized state to the drain (i.e., Γ

_{out,i}) [12,13]. Of course, the different relaxation rates internal to the localized state will also play a fundamental role in these experiments [12,13,15,25,45,46,47,48,49,50,51,52]. 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 [12,13,14,15], 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].

- (a)
- The first mechanism of error is the one that becomes important when the temperature of operation (T) of the single electron pump is energetically comparable to the charging energy (E
_{C}) of the confined states, i.e., E_{C}≈ K_{B}T, with the charging energy being the energy that the system must pay to increase by one electron the number of electrons in the localized state between the two gates, i.e., from N to N + 1. These thermal effects could lead to losing control of many electrons in each step of the cycle described in Figure 7. The probability of the occurrence of this type of errors is linked to the formula ${e}^{-\frac{{E}_{C}}{{K}_{B}T}}$ Hence E_{C}>> K_{B}T means that this probability is almost zero. As K_{B}T~24 meV for T = 300 K and ~0.24 meV for T = 3 K, the typical E_{C}~30–50 meV observed in single impurity/atom systems makes that these are immune to temperature errors even when they are operated above the liquid Helium temperatures (≥4.2 K). This explains why single-atom based single electron pumps (SAP’s) are extremely advantageous, as they do not need to be kept at ultra-low temperatures (<1 K) to operate in an environment completely immune from detrimental temperature effects/errors [12,13]. QD electron pumps are often limited in this sense as E_{C}for these systems is often limited to less than a few meV and as such require some complicated sub-kelvin temperature of operations to demonstrate their best performances [14]. However, as soon as the E_{C}of a QD increase to a value like the ones observed naturally in isolated single atom systems, for example by electrostatic confinements [14,47], these errors can also be suppressed in QDs electron pumps operating at temperatures ~4.2 K [14,15]. - (b)
- Another mechanism of errors that can affect single electron pumps, when they operate slightly above the 100 MHz frequencies of excitation, is the one related to non-adiabatic effects [45,48]. These errors are related to the poor efficiency in the achievement of the second step of the pumping cycle, as described in Figure 7b, i.e., the isolation step. This poor efficiency can be observed when the rates that control the back-tunneling of the electrons from the localized state back to the source, Γ
_{back}’s, allow the escape of the electrons to the source before the full isolation or before the emission to the drain [46,47,48]. This kind of error can particularly affect QD electron pumps as, for these systems, electrons are strongly affected when fast perturbations are exciting the system. At high frequencies of operations, these excitations, i.e., non-adiabatic excitations [48], 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].

_{relaxation}, 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 Γ
_{relaxation}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].

- (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 [15,50,51,52,53], but have also opened up the way to the use of the quantum pumping technology for novel quantum information schemes [45].

## 3. Conclusions

## Conflicts of Interest

## References

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**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].

**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, Γ

_{in,i}, and the tunneling barrier from state to the drain, Γ

_{out,i}. The leads (source/drain) represents an infinite reservoir of electrons with the all the possible spin (s

_{i}) and valley-orbital (p

_{i}) polarizations. The transport in the device can be controlled by applying a voltage to the V

_{gate}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 V

_{gate}terminal can also control the transparency of the tunneling barriers Γ

_{in,i}and Γ

_{out,}

_{i}. The V

_{Source-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.

**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., |V

_{Source-Drain}| < 5 mV, region between the one-electron (i.e., D

^{0}) 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].

**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 (B

_{C}) 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 > B

_{C}.

**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, T

_{C}) can be observed between 0 Tesla and 2 Tesla. This is most likely followed by a constant value of T

_{C}for any B > 2 Tesla.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Tettamanzi, G.C.
Unusual Quantum Transport Mechanisms in Silicon Nano-Devices. *Entropy* **2019**, *21*, 676.
https://doi.org/10.3390/e21070676

**AMA Style**

Tettamanzi GC.
Unusual Quantum Transport Mechanisms in Silicon Nano-Devices. *Entropy*. 2019; 21(7):676.
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**Chicago/Turabian Style**

Tettamanzi, Giuseppe Carlo.
2019. "Unusual Quantum Transport Mechanisms in Silicon Nano-Devices" *Entropy* 21, no. 7: 676.
https://doi.org/10.3390/e21070676