# Low-field electron mobility evaluation in silicon nanowire transistors using an extended hydrodynamic model

**Part of the following topical collections:**

## Abstract

Silicon nanowires (SiNWs) are quasi-one-dimensional structures in which electrons are spatially confined in two directions and they are free to move in the orthogonal direction. The subband decomposition and the electrostatic force field are obtained by solving the Schrödinger–Poisson coupled system. The electron transport along the free direction can be tackled using a hydrodynamic model, formulated by taking the moments of the multisubband Boltzmann equation. We shall introduce an extended hydrodynamic model where closure relations for the fluxes and production terms have been obtained by means of the Maximum Entropy Principle of Extended Thermodynamics, and in which the main scattering mechanisms such as those with phonons and surface roughness have been considered. By using this model, the low-field mobility of a Gate-All-Around SiNW transistor has been evaluated.

## Keywords

Nanowires Semiconductors Boltzmann equation Hydrodynamics## List of abbreviations

- EMA
Effective Mass Approximation

- GAA
Gate-All-Around

- MEP
Maximum Entropy Principle

- MC
Monte Carlo

- MBTE
Multiband Boltzmann Transport Equation

- NW
nanowire

- SP
Schrödinger–Poisson system

- Si
Silicon

- SiNW
Silicon nanowire

- SRS
Surface roughness scattering

- TB
Tight-Binding

## MSC

82D80 82D37 35Q20 75A15## 1 Introduction

In the last decades nanotechnologies made possible the production of innovative devices with promises of high density integration, for an exponential increasing of electronic systems complexity. Nanostructures and nanotechnologies are reaching important breakthroughs in single molecule sensing and manipulation, with fundamental applications. In particular, among these nanostructures, silicon nanowires (SiNW) are largely investigated for the central role assumed by Silicon (Si) in the semiconductor industry. Such device can be used as transistors [1, 2], logic devices [3], and thermoelectric coolers [4, 5], but also for other application fields such as biological and nanomechanical sensors [6, 7]. When the physical size of the system becomes smaller, quantum effects on electronic properties become important and then a description via quantum mechanics is required. These quantum effects arise in systems which confine electrons to regions comparable to their de Broglie wavelength.

In a nanowire (NW) the electronic states become subject to quantization in the two-dimensional transversal section, and the transport is due to the one-dimensional electron gas in the longitudinal dimension.

## 2 Methods

Charge transport in SiNWs, under reasonable hypotheses on the device’s dimensions, can be tackled using the 1-D Multiband Boltzmann Transport Equation (MBTE) coupled self-consistently with the 3-D Poisson and 2-D Schrödinger equations, in order to obtain the self-consistent potential and subband energies and wavefunctions. However, solving the MBTE numerically is not an easy task, because it forms an integro-differential system in two dimensions in the phase-space and one in time, with a complicate collisional operator. An alternative is to take the moments of the MBTE to obtain hydrodynamic-like models where the resulting system of balance equations can be closed by resorting to the Maximum Entropy Principle (MEP).

In the following we shall focus primarily on the mathematical method itself, whereas a minor emphasis will be given to the physical model, because some simplifications will be made which could lead to doubtful results.

## 3 Transport physics in SiNWs

In SiNWs the band structure is altered with respect to the bulk silicon, depending on the cross-section wire dimension, the atomic configuration, and the crystal orientation. Atomistic simulations are able to capture the nanowire band structure, including information about band coupling and mass variations as functions of quantization [8, 9, 10, 11, 12, 13]. In this paper we shall limit ourselves to the results obtained via the empirical *Tight-Binding* (TB) model [9].

*unprimed*valleys \(\Delta_{4}\) ([0 ± 10] and [00 ± 1] orthogonal to the wire axis) are projected into a unique valley in the

*Γ*point of the one-dimensional Brillouin zone. The subbands related to the

*primed*valleys \(\Delta_{2}\) ([±100] along the wire axis) are found at higher energies and exhibit a minimum located at \(k _{x}=\pm 0.37\pi /a _{0}\). The SiNW band gap, as well as the energy splitting between the \(\Delta_{2}\)–\(\Delta_{4}\) valleys increases with decreasing diameter of the nanowire. Moreover the subband isotropy break down at energies of the order of 150 meV above the (bulk) conduction-band maximum. From the energy dispersion relation \(E(k)\) obtained from the TB, one can evaluate the effective mass \(m ^{*}\) in the parabolic spherical band approximation. In this paper we shall consider the parameters obtained in [9] (see Table 1), which are valid for diameters greater then 3 nm. These values will certainly be affected by non-parabolic corrections.

Silicon nanowire constants

Symbol | Physical constant | Value |
---|---|---|

\(m_{e}\) | electron rest mass | 9.1095 × 10 |

| effective mass \(A =\Delta _{4}\) valley [9] | 0.27 \(m_{e}\) |

| effective mass \(B =\Delta _{2}\) valley [9] | 0.94 \(m_{e}\) |

\(T_{L}\) | lattice temperature | 300 K |

| mass density | 2.33 g/cm |

\(v_{s}\) | average sound speed | 9 × 10 |

\(D_{ac}\) | acoustic-phonon deformation potential | 9 eV |

\(D_{o}\) | intra-valley deformation potential g-scat [27] | 1.1 × 10 |

\(\sim \omega _{o}\) | intra-valley phonon energy [27] | 63.3 meV |

\(Z_{o}\) | number equivalent valleys [27] | 1 |

\(D_{iv}\) | inter-valley deformation potential f-scat [27] | 2 × 10 |

\(\sim \omega _{iv}\) | inter-valley phonon energy [27] | 47.48 meV |

\(Z_{iv}\) | number equivalent valleys [27] | 2 |

\(\varepsilon_{0_{A}}\) | \(A =\Delta _{4}\) valley energy minimum [9] | 0 |

\(\varepsilon_{0_{B}}\) | \(B =\Delta _{2}\) valley energy minimum [9] | 117 meV |

\(\Delta_{\mathrm{sr}}\) | rms height [27] | 0.3 nm |

\(\lambda_{\mathrm{sr}}\) | correlation length [27] | 1.5 nm |

The main quantum transport phenomena in SiNWs at room temperature, such as the source-to-drain tunneling, and the conductance fluctuation induced by the quantum interference, become significant only when the channel lengths are smaller than 10 nm [14]. For longer longitudinal lengths, which is the case we are going to simulate, semiclassical formulations based on the 1-D BTE can give reliable terminal characteristics when it is solved self-consistently by adding the Schrödinger–Poisson equations in the transversal direction.

*y*–

*z*plane by a SiO

_{2}layer which gives rise to a deep potential barrier having \(U= 3.2\mbox{ eV}\), and free to move in the orthogonal

*x*direction, having dimension \(L _{x}\) (see Fig. 1). Hence, it is natural to assume the following ansatz for the electron wave function

*μ*is the valley index (one \(\Delta_{4}\) valley and two \(\Delta_{2}\) valleys), \(l= 1\), \(N _{\mathrm{sub}}\) the subband index, \(\chi ^{\mu } _{l}(y,z)\) is the subband wave function of the \(\sqrt{ l}\)th subband and

*μ*th valley, and the term \(e ^{ \mathit{ik_{x}x}}/ L _{x}\) describes an independent plane wave in

*x*-direction, with wave-vector \(k _{x}\). The spatial confinement in the \((y,z)\) plane is governed by the Schrödinger–Poisson system (SP)

*Hartree approximation*). The electron density \(n[V ]\) is given by (2)

_{4}, where \(\rho_{l}^{\mu}(x, t)\) is the linear density in the

*μ*-valley and

*l*-subband which must be evaluated by the transport model (hydrodynamic/kinetic) in the free movement direction. We emphasize that the use of the effective mass approximation (2)

_{1}is probably valid for semiconductor nanowires down to 5 nm in diameter, below which atomistic electronic structure models need to be employed [15, 16]. The SP system forms a set of coupled nonlinear Partial Differential Equations, which are usually solved by an iteration between Poisson and Schrödinger equations. Since a simple iteration by itself does not converge, it is necessary to introduce an adaptive iteration scheme [17], where the Poisson equation has been solved by the finite-difference scheme proposed in [18], which can be used for every cross-section shape of the wire with complex geometries of the boundary/interface.

*η*th scattering rate is:

*η*th scattering rate. Phonon scattering has been tackled following the bulk Si scattering selection rules [12] whose details are given in [20]. But this is no more than a simplification, because major differences in the transport properties can appear including confined phonons [21] and anisotropic deformation potentials [22].

_{2}), and exchange-correlation energy (due to the electron–electron interaction) are neglected, which is reasonable for silicon thickness greater than 8 nm [25]. Moreover corner effects [26] have been neglected. In this case the SR scattering rate along the

*y*-direction is

_{2}interface

## 4 Extended hydrodynamic model

One of the most popular approaches is to solve the MBTE in a stochastic sense by Monte Carlo (MC) methods [21, 27, 28, 29] or by using deterministic numerical solvers [27, 30]. However, the extensive computations required by both methods as well as the noisy results obtained with MC simulations, make them impractical for device design on a regular basis.

*physics-based*hydrodynamic model is obtained, consistent with thermodynamics principles, valid in a larger neighborhood of local thermal equilibrium, and free of any tunable parameters.

## 5 Results and discussion

The main goal of this paper is to check if the above mentioned Extended hydrodynamic model is able to describe the quasi-equilibrium regime. Taking advantage of examples present in literature, we have considered a Gate-All-Around (GAA) SiNW transistor, with quadratic cross section. This is a Silicon nanowire with an added gate wrapped around it, in such a way we have a three contact device with source, drain, and gate. The device length is \(L _{x} = 120\) nm, the transversal dimensions \(L _{y}=L _{z} \leq 10\) nm, and the oxide thickness *tox* is 1 nm. The device is undoped, at room temperature and its cross sections are shown in Fig. 1.

*E*, i.e.,

- (i)
*equilibrium solution*

*ν*is the Fermi level, \(\varepsilon_{\mu}^{0}\) the valley energy minimum, and

*T*the electron temperature, which we shall assume to be the same in each subband and equal to the lattice temperature \(T _{L}\). The condition of zero net current requires that the Fermi level must be constant throughout the sample, and it can be determined by imposing that the total electron number equals the total donor number in the wire. Then, the linear electron density at equilibrium is:

- (ii)
*quasi-equilibrium solution*

- (iii)
*low-field mobility determination*

*x*-axis, we can skip the spatial dependence in the hydrodynamic model, which reduces to a system of Ordinary Differential Equations. The energies \(\varepsilon^{\mu } _{lx}\) and wave functions \(\chi^{ \mu } _{lx}\) for each subband are imported from the previous steps (and kept fixed), as well as the linear density (15) which is used as initial condition. The other initial conditions are

About step (iii), the stationary regime of the hydrodynamic system has been reached in some ps, and the CPU effort varies according to the voltage \(V _{G}\) with a maximum of one hour.

_{4}in the cross section \(x= 60\) nm, perpendicular to the transport direction, is shown in the Figs. 7, 8, 9 for \(V _{G} = 0.16,0.6,1\) V respectively. For small gate voltage, the volume charge is peaked in the center of the wire as shown in Fig. 7. As the gate voltage increases, the electron density is peaked close to the oxide interface (see Figs. 8 and 9). This phenomenon can be seen also in Fig. 10 where we plot the electron density (2)

_{4}and total potential \(V _{\mathrm{tot}}\) in the cross section \(y= 0\) nm and \(x= 60\) nm, for \(V _{G} = 0.6\) V. In particular one can observe the effect of the wave function penetration in the oxide and the formation of a surface inversion layer, similar to a usual MOSFET channel.

The presented results have been obtained using MATLAB running in an AMD Phenom II X6 1090T 3.2 GHz and 8 Gb RAM.

## 6 Conclusions

We present a theoretical study of low-field electron mobility in a Gate-All-Around silicon nanowires, having rectangular cross section, based on a hydrodynamic model coupled to the Schrödinger–Poisson equations. The hydrodynamic model has been formulated by taking the moments of the multisubband Boltzmann equation, and by closing the obtained hierarchy of balance equations with the use of the Maximum Entropy Principle. The most relevant scattering mechanisms, such as scattering of electrons with acoustic and non-polar optical phonons and surface roughness, have been included. The results show a good qualitative agreement with data available from the literature, confirming that this hydrodynamic model is valid in the quasiequilibrium regime limit. The study of off-equilibrium transport phenomena as well as of thermoelectric effects for such structures, using also circular cross-sections of the wire, will be the subjects of future researches.

## Notes

### Acknowledgements

We acknowledge the support of the project “Modellistica, simulazione e ottimizzazione del trasporto di cariche in strutture a bassa dimensionalità”, Università degli Studi di Catania—Piano della Ricerca 2016/2018 Linea di intervento 2.

### Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

### Authors’ contributions

All authors have jointly worked to the manuscript with an equal contribution. All authors read and approved the final manuscript.

### Funding

This research has been supported by Università degli Studi di Catania.

### Competing interests

The authors declare that they have no competing interests.

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