# Variable-Barrier Quantum Coulomb Blockade Effect in Nanoscale Transistors

^{1}

^{2}

^{3}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Devices

_{SOI}≈ 10 nm and a gate oxide with a thickness t

_{ox}≈ 10 nm, while the source, drain and gate electrodes are formed by aluminum contact.

_{D}≈ 1 × 10

^{18}cm

^{−3}) in the channel region, along with the source and drain leads. Similar doping condition is valid for the source and drain leads for Device-A. The potential wells induced by several ionized P−donors in the channel are schematically illustrated in Figure 1d. Considering the channel dimensions and dopant concentration, it is estimated that around 5 P−donors are present in the channel (labeled as Pi, with i = 1–5).

#### 2.2. Device Configuration for Theoretical Calculations

_{DS}and the gate voltage V

_{G}. Both reservoirs and gate electrode are capacitively connected to the QD through capacitances C

_{S}, C

_{D}, and C

_{G}, respectively, with the total capacitance of the system being: C

_{∑}= C

_{S}+ C

_{D}+ C

_{G}. In this device, the transfer of electrons from reservoirs to the QD or vice versa is governed mainly by the potential differences between the leads and the QD. We chose the reference electrostatic potential in such a way that the energy levels in the QD are independent of the bias voltages [32]. On the contrary, the Fermi energies of the leads are described as a function of the different capacitances and applied voltages as:

_{B}T) is much smaller than the level spacing (Δ) of the QD, while this level spacing itself must be smaller than the charging energy (E

_{C}= e

^{2}/2C

_{∑}) of the QD; (b) tunnel resistance (R

_{t}) of both barriers is greater than quantum resistance (h/e

^{2}= 25.81 kΩ) which ensures suppression of the higher-order tunneling processes; (c) a continuum of states in both electron reservoirs is assumed, ensuring the absence of discreteness in the local density of states (LDOS) of the leads [1]. Moreover, all types of internal relaxations and electron-electron interactions within the QD are also neglected in this model.

#### 2.3. Theoretical Formalism

_{T}, while the Hamiltonians for an ideal, isolated QD and for the source (drain) reservoirs are H

_{Dot}and H

_{S(D)}, respectively. Here:

_{e}into the QD.

_{j}= either ‘0′ or ‘1′) and energy redefined as ${\tilde{\epsilon}}_{\mathrm{j}}={\epsilon}_{\mathrm{j}}+{E}_{\mathrm{C}}\left(\mathrm{eV}\right)$, connected to both the reservoirs. Here, ${\Gamma}_{\mathrm{j}}^{\mathrm{S}\text{}\left(\mathrm{D}\right)}$ is the bare tunneling rate of the respective energy level coupled with source (drain), whereas ‘$f\left(x\right)$’ defines the Fermi function at the temperature T:

^{−1}.

- (i)
- For infinitely high tunnel barriers, it would suffice to consider ${\Gamma}_{\mathrm{j}}^{\mathrm{S}\text{}\left(\mathrm{D}\right)}$= ${\mathsf{\Gamma}}^{\mathrm{S}\left(\mathrm{D}\right)}$ = constant.
- (ii)
- For finite and bias-dependent barrier, the bare tunneling rate is varying with the bias voltage as presented below:

## 3. Results and Discussion

#### 3.1. Experimental Evidences for the Effect of Variable Tunnel Barriers of a QD

_{1}and QD

_{2}in Figure 1a and as broader potential wells in Figure 1b. I

_{D}−V

_{G}characteristics measured at low-temperature (5 K) for Device-A are presented in Figure 3a. In this figure, SET current peaks can be identified, labeled as a

_{1}, a

_{2}, a

_{3}, a

_{4}, b

_{1}, and b

_{2}. Each of these current peaks has several associated sub-peaks. For a

_{1}–a

_{4}peaks, the gap between consecutive subpeaks (Δ

_{1}) is ~22 ± 3 mV, while for b

_{1}, b

_{2}peaks, the same gap (Δ

_{2}) is ~50 ± 2 mV. These associated subpeaks are most likely due to transport mediated by the excited states of their respective QD [42]. Hence, it is reasonable to assume that the a

_{1}–a

_{4}peaks are associated with QD

_{1,}while the b

_{1}and b

_{2}peaks are associated with QD

_{2}. The gap between a

_{1}and a

_{2}(E

_{C1}) is 139 ± 3 mV. Similar gap is observed between a

_{3}and a

_{4}. The energy gap between a

_{2}and a

_{3}is 160 ± 2 mV, which is also the sum of E

_{C1}and Δ

_{1}. This is the clear indication of quantum Coulomb blockade phenomenon.

_{1}toward a

_{4}peaks, which strongly suggests gradual increment of the tunnel rate with increasing gate voltage. The relation between a

_{1}–a

_{4}peaks is schematically presented in Figure 3b in correlation with a simplified representation of transport and electrical characteristics. This simplified picture is emphasizing the expected behavior under the observation of variable-barrier quantum Coulomb blockade in the QD

_{1}system.

_{D}−V

_{G}characteristic of Device-B measured at T = 6 K is presented in Figure 3c. The device configuration of Device-B is basically uniformly doped MOSFET in SOI configuration as presented in Figure 1d. Five single-electron-current peaks are observed with irregular spacing before the onset of FET current. These current peaks are separated by Coulomb energy. The spacing between these current peaks are irregular, which generally originated from different quantum dots. Considering the devices configuration of the Device-B, these quantum dots are most likely due to donor present (P−donor in this case) in the channel region of the device as reported earlier [37,38,39,43]. Due to the different positions of the donor atoms in the channel region, all donors have different barrier parameters and that can be controlled by the gate voltage. The origin of five SET peaks can be directly correlated to the existence of 5 P−donors in the channel region of the device as estimated from the device designing. The schematic dopant distribution and potential configuration of this device structure are shown in Figure 1c,d. We also observed transport through the excited state of the donor QD with the average separation of excited state from the ground state of the donor as 8 ± 2 mV. This separation is tentatively consisted with the energy spectrum of the P−donor [44]. In addition, we also observed that the heights of the current peaks are gradually enhanced with the increasing gate voltage. This suggests that the tunneling rates are also tuned by the gate voltage even in the case of donor-induced QDs.

#### 3.2. Numerical Analysis of Electron Transport

**i.**- Electron Transport through Two Energy Levels:

_{0}, P

_{1}, and P

_{2}are the occupation probabilities of respective electronic configurations (0,0), (1,0), and (0,1) as schematically depicted in Figure 4b. These are expressed as:

_{C}= 10.66 meV for numerical calculation of the characteristics of the device. All these calculations are performed at low temperature of T = 4 K, comparable to the condition for the experimental data. Here, we accounted for both situations: (i) the tunneling rate is constant considering the infinite barrier height and (ii) tunneling rate is varying with the applied gate voltage. For the first case, the calculated I

_{D}-V

_{G}characteristic for different charge states is plotted in Figure 4d. Two separate current sub-peaks within a peak can be assigned to SET transport involving the ground state and the 1st excited state. The level separation of the sub-peaks, Δ, and an alternative energy separation of E

_{C}and E

_{C}+ Δ are clear signatures of quantum Coulomb blockade. The stability diagram (i.e., the plot of I

_{D}in the V

_{G}−V

_{DS}plane) is shown in Figure 4e, where the excited-state features are also clearly observed as marked by white arrows in the first charge state. Successive incorporation of charges in the device is visible in the stability diagram as N

_{e}changes from 0 to 4.

_{D}| in V

_{G}−V

_{DS}plane (Figure 4g). The systematic enhancement of the current intensity of conducting region of the stability diagram is clearly visible when N

_{e}changes from 0 to 4, consistent with the recent experimental observations.

**ii.**- Electron Transport through Three Energy Levels:

_{D}-V

_{G}features for constant- and variable-height tunnel barriers are presented in Figure 5a,b, respectively. In each SET current peak, we observed three subpeaks as expected due to the accessibility of three energy levels in the bias window. The realistic device feature for the variable-height barrier case is also clearly observed in the Figure 5b. The stability diagram corresponding to infinite- and variable-height barrier cases are presented in Figure 5c,d, respectively. The systematic incorporation of additional energy levels in the transport path is depicted by white arrows in Figure 5c,d. The features of quantum Coulomb blockade and the differences between infinite- and variable-height barrier configurations are clearly visible in these figures, confirming the feasibility of our approach towards the qualitative replication of the experimental data.

## 4. Conclusions

_{C}and E

_{C}+ Δ correlated to spin degeneracy of the energy levels along with the modification of tunnel rate due to variation in the tunnel barrier. To qualitatively reproduce the experimental findings of realistic devices, we have numerically calculated the current voltage characteristics for the constant and variable tunnel barrier conditions. We showed that the numerical results for QD with two and three levels accessible for tunneling transport. The modified theoretical formalism closely replicates the nano-scaled SET devices fabricated in two-dimensional electron gas (2DEG) systems, semiconductor QDs, and dopants as QDs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**,

**b**) Schematic device structure and schematic potential configuration of Device-A (with a nominally undoped nanoscale channel). (

**c**,

**d**) Schematic device structure and schematic potential configuration of Device-B (with uniformly doped channel).

**Figure 2.**(

**a**) Schematic potential diagram of a double-barrier QD system with the tunnel barriers also controlled by the gate voltage. (

**b**) Equivalent electrical circuit model of a QD in the double-barrier configuration with symmetric bias.

**Figure 3.**Device A (

**a**) I

_{D}−V

_{G}characteristics measured at T = 5 K, depicting also the single-electron-tunneling current peaks a

_{1}–a

_{4}and b

_{1}, b

_{2}, likely originated from QD

_{1}and QD

_{2}, respectively. (

**b**) Schematic summarization of tunneling transport through the QD

_{1}, suggesting the variable-barrier QCB. Device B (

**c**) I

_{D}−V

_{G}plot shows the SET transport through isolated P−donors in uniformly-doped SOI-FET channels.

**Figure 4.**(

**a**) Schematic representation of a QD with two energy levels accessible in the bias window, connected with source and drain reservoirs. (

**b**) Probable transitions from $\left|\mathsf{\varphi}\right.\u232a$ (P

_{0}) to $\left|\mathsf{\psi}\right.\u232a$(P

_{1}and P

_{2}) are shown. (

**c**) Schematic representations of the successive incorporation of electrons in the QD for the quantum Coulomb blockade case. (

**d**,

**e**) I

_{D}−V

_{G}characteristics and stability diagram of a SET system with constant bare tunnel rates, respectively. (

**f**,

**g**) I

_{D}−V

_{G}characteristics and stability diagram of a practical SET setup with biasing-dependent tunnel rates, respectively. Arrows indicate the onset of transport through a new energy level.

**Figure 5.**(

**a**,

**b**) Simulated I

_{D}−V

_{G}characteristics with three energy levels within the bias window for infinite- and variable-height barrier devices, respectively. (

**c**,

**d**) Simulated stability diagram for the same device configurations. Arrows indicate the onset of transport through a new energy level.

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**MDPI and ACS Style**

Yadav, P.; Chakraborty, S.; Moraru, D.; Samanta, A. Variable-Barrier Quantum Coulomb Blockade Effect in Nanoscale Transistors. *Nanomaterials* **2022**, *12*, 4437.
https://doi.org/10.3390/nano12244437

**AMA Style**

Yadav P, Chakraborty S, Moraru D, Samanta A. Variable-Barrier Quantum Coulomb Blockade Effect in Nanoscale Transistors. *Nanomaterials*. 2022; 12(24):4437.
https://doi.org/10.3390/nano12244437

**Chicago/Turabian Style**

Yadav, Pooja, Soumya Chakraborty, Daniel Moraru, and Arup Samanta. 2022. "Variable-Barrier Quantum Coulomb Blockade Effect in Nanoscale Transistors" *Nanomaterials* 12, no. 24: 4437.
https://doi.org/10.3390/nano12244437