# Quantum Treatment of Inelastic Interactions for the Modeling of Nanowire Field-Effect Transistors

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## Abstract

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## 1. Introduction

## 2. General Theoretical Framework

#### 2.1. Dyson Equation

#### 2.2. Self-Consistent Born Approximation

#### 2.3. Lowest Order Approximation

Algorithm 1Nth-order LOA calculation. |

$N\leftarrow $ <the order of LOA>for $i\leftarrow 0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}to\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}N$ doif $i=0$ then${g}_{0}^{r}\leftarrow \left[EI-H-{\Sigma}_{C}^{r}\right]$, E (Energy), H (Hamiltonian), and ${\Sigma}_{C}$ (Contact self-energy) ${g}_{0}^{\lessgtr}\leftarrow {g}_{0}^{r}{\Sigma}_{C}^{\lessgtr}{g}_{0}^{a}$, the superscript r (Retarded), a (Advanced), and ≶ (Lesser/Greater) $\Delta {g}_{0}^{\lessgtr}\leftarrow {g}_{0}^{\lessgtr},\Delta {g}_{0}^{r}\leftarrow {g}_{0}^{r}$ elsefor $n\leftarrow 0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}to\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}i-1$ do: interacting self energy calculation ${\Sigma}_{int,\phantom{\rule{0.166667em}{0ex}}i}^{\lessgtr}$ and ${\Sigma}_{int,\phantom{\rule{0.166667em}{0ex}}i}^{r}$ : perturbation term calculation $temp(\Delta {g}_{i}^{r})\leftarrow {g}_{0}^{r}{\Sigma}_{int,\phantom{\rule{0.166667em}{0ex}}i-n}^{r}\Delta {g}_{n}^{r}$ ⇓ applying Langreth Theorem $temp(\Delta {g}_{i}^{\lessgtr})\leftarrow {g}_{0}^{r}{\Sigma}_{int,\phantom{\rule{0.166667em}{0ex}}i-n}^{r}\Delta {g}_{n}^{\lessgtr}+{g}_{0}^{r}{\Sigma}_{int,\phantom{\rule{0.166667em}{0ex}}i-n}^{\lessgtr}\Delta {g}_{n}^{a}+{g}_{0}^{\lessgtr}{\Sigma}_{int,\phantom{\rule{0.166667em}{0ex}}i-n}^{a}\Delta {g}_{n}^{a}$ $\Delta {g}_{i}^{r}\leftarrow \Delta {g}_{i}^{r}+temp(\Delta {g}_{i}^{r})$ $\Delta {g}_{i}^{\lessgtr}\leftarrow \Delta {g}_{i}^{\lessgtr}+temp(\Delta {g}_{i}^{\lessgtr})$ end for${g}_{i}^{r}\leftarrow {g}_{i-1}^{r}+\Delta {g}_{i}^{r}$ ${g}_{i}^{\lessgtr}\leftarrow {g}_{i-1}^{\lessgtr}+\Delta {g}_{i}^{\lessgtr}$ end ifend for |

#### 2.4. Rescaling Technique

#### 2.5. Matrix Form of the Padé Approximants

#### 2.6. Richardson Extrapolation

## 3. Applications to Electron and Phonon Transports in a Nanowire Transistor

#### 3.1. Electron–Phonon Scattering in a Nanowire Transistor

#### 3.2. Anharmonic Phonon–Phonon Scattering in A Nanowire

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Feynman diagrams for the Dyson equation showing the relation between the noninteracting electron Green’s function ${g}_{0}$ (thin lines with an arrow) and fully interacting one G (thick lines with an arrow): The dashed line denotes the free phonon propagator.

**Figure 2.**Feynman diagrams for self-consistent Born approximation (SCBA) Green’s functions, (

**a**) ${G}_{1}$ (at first iteration) and (

**b**) ${G}_{2}$ (at second iteration): Conserving (red-underlined) and non-conserving (black-underlined) terms are arranged according to ascending interaction order. Noninteracting electron Green’s functions are described by thin lines with an arrow while free phonon propagators are represented by dashed lines.

**Figure 3.**Feynman diagrams for Lowest Order Approximation (LOA) Green’s functions: (

**a**) ${g}_{1LOA}$ at the first order and (

**b**) ${g}_{2LOA}$ at the second order in interactions. Conserving terms are red-underlined. Noninteracting electron Green’s functions are described by thin lines with an arrow while free phonon propagators are represented by dashed lines.

**Figure 4.**Schematic view of a Si gate-all-around (GAA) nanowire (NW) field-effect transistor (FET) crystallographically oriented along the $\langle 100\rangle $ direction with a ${H}_{NW}\left(3\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}\right)\times {W}_{NW}\left(3\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}\right)$ square cross section: (Electron transport) The gate length is ${L}_{G}=13$ nm, and the length of source/drain region measures ${L}_{S/D}=10$ nm. The NW structure is surrounded by a 1-nm-thick silicon dioxide layer. The concentration of donors in the source and drain regions is ${N}_{S/D}=1\times {10}^{20}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$. (Phonon transport) The NW total length (${L}_{G}+{L}_{S}+{L}_{D}$) is 60 nm. The NW is undoped, ungated, and free of the oxide layer.

**Figure 5.**(

**a**) Electronic conduction band-structure of the Si NW sketched in Figure 4, obtained with a full band tight-binding sp${}^{3}$d${}^{5}$s${}^{*}$ model without spin-orbit coupling. (

**b**) Phonon dispersion relation obtained with a modified valence-force-field method. ${L}_{x}$ is a slab length.

**Figure 6.**${I}_{D}-{V}_{G}$ transfer characteristics of the n-type 3 nm × 3 nm square cross-sectional Si GAA NW-FET obtained for ballistic regime and by SCBA, 1st-order LOA, Padé 0/1, and Padé 1/2.

**Figure 7.**Thermal currents at room temperature in the 3 nm × 3 nm square cross-sectional Si NW of Figure 4 obtained for the ballistic regime and within the SCBA, Padé 0/1, Padé 1/1, Padé 1/2, and the first-order Richardson extrapolation.

**Table 1.**Accuracy ($\epsilon =100\times |{\mathcal{I}}_{SCBA}-\mathcal{I}|/{\mathcal{I}}_{SCBA}$ where $\mathcal{I}$ is ballistic, LOA, Padé, or Richardson current) and efficiency (# of iterations) comparisons of 1st and 3rd LOA, Padé 0/1 and 1/2, and the corresponding Richardson currents with ballistic and SCBA currents at ${V}_{G}=0.6$ $\mathrm{V}$.

Ballistic | LOA1 | LOA3 | Padé 0/1 | Padé 1/2 | Richardson | SCBA | |
---|---|---|---|---|---|---|---|

Current [A] | 8.9 × 10${}^{-6}$ | 3.57 × 10${}^{-6}$ | −2.616 | 5.6 × 10${}^{-6}$ | 5.9 × 10${}^{-6}$ | 6.1 × 10${}^{-6}$ | 6.3 × 10${}^{-6}$ |

$\epsilon $ [%] | 42.1 | 42.0 | 4.2e7 | 10.8 | 6.4 | 2.0 | 0.0 |

Number of iterations | 0 | 1 | 6 | 1 | 6 | 6 | 35 |

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**MDPI and ACS Style**

Lee, Y.; Logoteta, D.; Cavassilas, N.; Lannoo, M.; Luisier, M.; Bescond, M. Quantum Treatment of Inelastic Interactions for the Modeling of Nanowire Field-Effect Transistors. *Materials* **2020**, *13*, 60.
https://doi.org/10.3390/ma13010060

**AMA Style**

Lee Y, Logoteta D, Cavassilas N, Lannoo M, Luisier M, Bescond M. Quantum Treatment of Inelastic Interactions for the Modeling of Nanowire Field-Effect Transistors. *Materials*. 2020; 13(1):60.
https://doi.org/10.3390/ma13010060

**Chicago/Turabian Style**

Lee, Youseung, Demetrio Logoteta, Nicolas Cavassilas, Michel Lannoo, Mathieu Luisier, and Marc Bescond. 2020. "Quantum Treatment of Inelastic Interactions for the Modeling of Nanowire Field-Effect Transistors" *Materials* 13, no. 1: 60.
https://doi.org/10.3390/ma13010060