# Study of Electronic and Transport Properties in Double-Barrier Resonant Tunneling Systems

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## Abstract

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## 1. Introduction

## 2. Theoretical Model

^{®}) [42]. As a starting point, we must consider the effect on the potential of the donor density and electron density in the system—this can be modeled by means of the Poisson equation,

#### 2.1. A Device Macroscopically Large in the Transverse Directions

#### 2.2. Cut-Off Frequency Calculation

## 3. Results and Discussion

#### Comparison with Experimental Data

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Scheme of the resonant tunneling diode (RTD), with doping ${n}_{d}$ in the outer regions, two Al${}_{0.3}$Ga${}_{0.7}$As barriers, a GaAs well, and two outer regions of the GaAs undoped with two metal contacts in the external regions.

**Figure 2.**Conduction band profile. The dashed line correspond to the quasi-Fermi Level. The calculations are for ${L}_{w}=4$ nm, ${L}_{b}=3$ nm, ${L}_{s}=3$ nm, ${L}_{d}=12$ nm and ${n}_{d}=1.2\times {10}^{18}$ cm${}^{-3}$.

**Figure 3.**Potential energy for the system in equilibrium (bias voltage 0.0 V), the blue curve corresponds to the probability density of the resonant state, and the red dashed curve is the energy for this state ${E}_{0}$. The quasi-Fermi level is also presented with the blue dashed curve. The calculations are for ${L}_{w}=4$ nm, ${L}_{b}=3$ nm, ${L}_{s}=3$ nm, ${L}_{d}=12$ nm and ${n}_{d}=1.2\times {10}^{18}$ cm${}^{-3}$.

**Figure 4.**Potential energy change with bias voltage from 0.0 V to 0.4 V, the blue curve corresponds to the resonant state probability density and the red curve is the energy for this state ${E}_{0}$. The quasi-Fermi level is also presented by the dark blue dashed line for emitter and collector. The calculations are for ${L}_{w}=4$ nm, ${L}_{b}=3$ nm, ${L}_{s}=3$ nm, ${L}_{d}=12$ nm, and ${n}_{d}=1.2\times {10}^{18}$ cm${}^{-3}$.

**Figure 5.**Transmission coefficient for different values of bias voltage, the black curve is for ${L}_{w}$ = 4 nm and, the red curve is for ${L}_{w}$ = 10 nm. (

**a**) with ${n}_{d}$ fixed at 1.2$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$] and (

**b**) with ${n}_{d}$ fixed at 10$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$]. The shaded area indicates the region between the bottom of the conduction band and the quasi-Fermi level at the emitter. As indicated by the arrow in (

**b**), the voltage for each curve varies from 0.0 V to 0.6 V in steps of 0.05 V.

**Figure 6.**Transmission coefficient for different values of ${L}_{w}$, the red curve corresponds to 0.0 V, and the black curve corresponds to 0.4 V. (

**a**) with ${n}_{d}$ fixed at 1.2$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$], and (

**b**) with ${n}_{d}$ fixed at 10$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$]. The shaded area indicates the region between the bottom of the conduction band and the quasi-Fermi level at the emitter.

**Figure 7.**Tunneling current density for two different values of ${L}_{w}$ as a function of bias voltage, in (

**a**) with ${n}_{d}$ = 1.2$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$], and (

**b**) with ${n}_{d}$ = 10$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$]. Figure (

**c**) shows the transmission for three different values of the Al concentration in the barriers, x = 0.2, 0.3, and 0.4 for a system of three regions Al${}_{x}$Ga${}_{1-x}$As/GaAs/Al${}_{x}$Ga${}_{1-x}$As. The inset shows the current density for these three systems taking ${L}_{w}$ = 4 nm and ${L}_{b}$ = 3 nm. In figures (

**a**,

**b**), the cut-off frequencies have been included for all the arrangements calculated (black text corresponds to ${L}_{w}$ = 4 nm and red text corresponds to ${L}_{w}$ = 10 nm) by taking two different values of ${\tau}_{0}$, 0.1 ps and 0.2 ps.

**Figure 8.**(

**a**) Conductance for ${L}_{w}=4$ nm, for two different donor concentrations in units of ${G}_{0}={e}^{2}/\pi {\hslash}^{2}$, solid black line ${n}_{d}$ = 1.2$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$], and dashed red line ${n}_{d}$ = 10$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$]. (

**b**) Corresponding self-consistent potentials. The curves were calculated at $T=5$ K.

**Figure 9.**(

**a**) Conductance for ${L}_{w}=10$ nm, for two different donor concentrations in units of ${G}_{0}={e}^{2}/\pi {\hslash}^{2}$, solid black line ${n}_{d}$ = 1.2$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$], and dashed red line ${n}_{d}$ = 10$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{18}$ [1/cm${}^{3}$]. (

**b**) Corresponding self-consistent potentials. The curves were calculated at $T=5$ K.

**Figure 10.**RTD structure composed of 9 layers that are expanded in detail in Table 3. DBRTD stands for Double-Barrier Resonant Tunneling Diode.

**Figure 11.**Self-consistent potential corresponding to the conduction band obtained numerically with the experimental parameters detailed in Table 3.

**Table 1.**Energy associated with the conductance peaks and potential at the center of the well and their differences $\Delta E$ for the two calculated concentrations, the data correspond to Figure 8.

${\mathit{n}}_{\mathit{d}}$ (${10}^{18}$ [1/cm${}^{3}$]) | 1.2 | 10 | $\mathsf{\Delta}\mathit{E}$ (10${}^{-3}$ eV) |
---|---|---|---|

V (eV) | 0.0105 | 0.0193 | 8.8 |

${E}_{1}$ (eV) | 0.1213 | 0.1299 | 8.6 |

**Table 2.**Energy associated with the conductance peaks and potential at the center of the well and their differences $\Delta E$ for the two calculated concentrations, the data correspond to Figure 9.

${\mathit{n}}_{\mathit{d}}$ (${10}^{18}$ [1/cm${}^{3}$]) | 1.2 | 10 | $\mathsf{\Delta}\mathit{E}$ (10${}^{-3}$ eV) |
---|---|---|---|

V (eV) | 0.0138 | 0.0235 | 9.7 |

${E}_{1}$ (eV) | 0.0463 | 0.0558 | 9.5 |

${E}_{2}$ (eV) | 0.1432 | 0.1523 | 9.1 |

${E}_{3}$ (eV) | 0.2986 | 0.3076 | 9.0 |

**Table 3.**Parameters corresponding to each of the layers in Figure 10.

Parameters by Layer | |||
---|---|---|---|

Layer | Material | Dimensions (nm) | Doping (${\mathit{n}}^{+}$cm${}^{-\mathbf{3}}$) |

1 | In${}_{0.53}$Ga${}_{0.47}$As | 400 | 1 × 10${}^{19}$ |

2 | In${}_{0.53}$Ga${}_{0.47}$As | 25 | 3 × 10${}^{18}$ |

3 | In${}_{0.53}$Ga${}_{0.47}$As | 5 | |

4 | AlAs | 1.1 | |

5 | In${}_{0.8}$Ga${}_{0.2}$As | 3.5 | |

6 | AlAs | 1.1 | |

7 | In${}_{0.53}$Ga${}_{0.47}$As | 5 | |

8 | In${}_{0.53}$Ga${}_{0.47}$As | 25 | 3 × 10${}^{18}$ |

9 | In${}_{0.53}$Ga${}_{0.47}$As | 45 | 2 × 10${}^{19}$ |

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**MDPI and ACS Style**

Gil-Corrales, J.A.; Vinasco, J.A.; Mora-Ramos, M.E.; Morales, A.L.; Duque, C.A.
Study of Electronic and Transport Properties in Double-Barrier Resonant Tunneling Systems. *Nanomaterials* **2022**, *12*, 1714.
https://doi.org/10.3390/nano12101714

**AMA Style**

Gil-Corrales JA, Vinasco JA, Mora-Ramos ME, Morales AL, Duque CA.
Study of Electronic and Transport Properties in Double-Barrier Resonant Tunneling Systems. *Nanomaterials*. 2022; 12(10):1714.
https://doi.org/10.3390/nano12101714

**Chicago/Turabian Style**

Gil-Corrales, John A., Juan A. Vinasco, Miguel E. Mora-Ramos, Alvaro L. Morales, and Carlos A. Duque.
2022. "Study of Electronic and Transport Properties in Double-Barrier Resonant Tunneling Systems" *Nanomaterials* 12, no. 10: 1714.
https://doi.org/10.3390/nano12101714