Active and Passive Electronic Components

Volume 2016, Article ID 6068171, 8 pages

http://dx.doi.org/10.1155/2016/6068171

## Impact of Band Nonparabolicity on Threshold Voltage of Nanoscale SOI MOSFET

ORDIST, Grad. School of Sci. & Eng., Kansai University, 3-3-35 Yamate-cho, Suita, Osaka 564-8680, Japan

Received 24 September 2016; Revised 13 November 2016; Accepted 16 November 2016

Academic Editor: Mingxiang Wang

Copyright © 2016 Yasuhisa Omura. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper reconsiders the mathematical formulation of the conventional nonparabolic band model and proposes a model of the effective mass of conduction band electrons including the nonparabolicity of the conduction band. It is demonstrated that this model produces realistic results for a sub-10-nm-thick Si layer surrounded by an SiO_{2} layer. The major part of the discussion is focused on the low-dimensional electron system confined with insulator barriers. To examine the feasibility of our consideration, the model is applied to the threshold voltage of nanoscale SOI FinFETs and compared to prior experimental results. This paper also addresses a model of the effective mass of valence band holes assuming the nonparabolic condition.

#### 1. Introduction

In the last 3 decades, silicon-on-insulator (SOI) MOSFETs have been attracting attention because of their high short-channel effect immunity [1] and excellent potential with regard to future nanoscale devices [2]. Therefore, many studies on quantum confinement effects have been performed. The author predicted that the threshold voltage () of ultrathin-body (UTB) SOI MOSFET rises as SOI body thickness () is reduced due to the quantum-mechanical confinement effect [3]. Several studies have demonstrated experiments [4] that, they believe, demonstrate the effect of quantum confinement. However, the author demonstrated that the apparent rise of of UTB SOI MOSFETs is due to more than the quantum effect; it includes the semiclassical effect [5]. Other experiments strongly suggested that a simple parabolic band model is not appropriate in analyzing of nanoscale SOI MOSFETs [6, 7]; in these experiments, the threshold voltage was lower than the simulation values that assume the parabolic conduction band ( bands for Si) for nm. The possibility of determining the limits of the conventional parabolic band approximation is a motivation of this study. Recently, a couple of research articles demonstrated the importance of band nonparabolicity in the analysis of the transport characteristics of nanoscale materials [8, 9], where the first principle calculation and the tight-binding method are used to compute the electronic states. In practical situations, however, demand an analytical closed form for the quantized energy levels and an effective mass tensor is requested for electronic device designs, since they greatly reduce the time costs.

This paper reconsiders the mathematical formulation of the conventional nonparabolic band model. This paper examines whether some perturbations can be added to the conventional model for convenience. In the following discussion, this paper focuses on a low-dimensional electron system confined by insulator barriers. We discuss the impact of the nonparabolic conduction band in Si on the effective mass and propose an analytical expression for the effective mass of electrons including the conduction band nonparabolicity. The model is applied to the threshold voltage of nanoscale SOI FinFETs, and its validity is examined. By examining the mathematical basis for the effective mass of electrons that have conduction band nonparabolicity, this paper also illuminates a model for the effective mass of holes having valence band nonparabolicity.

#### 2. Modeling Nonparabolic Band Structure

When an isotropic band structure is assumed, it is conventionally known that its form can be expressed generally as [10]where is the effective mass of electrons in the isotropic band, is the nonparabolic band factor, is the energy of electrons in the nonparabolic band, and is the electron energy expression in the form of the parabolic band scheme. Here we reconsider how the realistic effective mass for device analysis should be estimated; the above simplified nonparabolic band model is assumed for the conduction band electrons of a three-dimensional quasi-free electron system.

When an external field effect is taken account of, we must examine whether the following formulation is theoretically valid or not:where is the perturbation energy corresponding to the 1st-order perturbation generated by the external electric field and is the perturbation factor. When (2) is valid, (2) can be rewritten asIn order to examine the availability of (3), the right-hand side of (3) is changed into Tayler’s power series. where is the expansion coefficient and . If (4) is meaningful, should be the eigenvalue of the corresponding Schrödinger’s equation. That is, (4) should be equivalent to the following operator representation:where all suffice and correspond to those in (4) and is the perturbation term that is expressed asWhen we can assume that the amplitude of is smaller than that of [11], we approximately haveEquation (7) has the same form as (3). However, as the above formulation indicates, the meaning of (7) differs from that of (3) because (7) has been examined with the Hamiltonian operator representation in spite of the approximations used. Provided that prominent nonlinear effects are not significant, it can be concluded that (7) holds important physical meaning in terms of evaluating nonparabolic band effects on transport characteristics.

For the two-dimensional electron system, (7) can be rewritten as [11]for the th subband attributed to the specific conduction band, where is the subband energy level of electrons, is the discrete wavenumber of the subband labeled “” in the quantum well, and is the energy component of the transport direction. Equation (8) is valid only for based on the same assumption used for (7). Actually, it is easily found that is much smaller than for electric fields less than 10^{7} V/cm and < 10 nm [10]; therefore, (8) is applicable to the analysis of nanoscale MOSFET characteristics.

The result described above yields an expression for the effective mass tensor () for two-dimensional electron systems around the subband bottom as in [14]. Before using (8) for direct calculations, we review the conventional idea of the effective mass. We must recall the fact that electronic states with finite dimensions are inherently discrete; the difference from a very small size material is just the magnitude of between adjacent states. Naive calculation discards the term of the group velocity, but this is erroneous in terms of physics. The correct calculation result is given aswhere “*j*” means confinement direction and is the “*effective group velocity*” along the direction labeled by “*j*.” Given that label “*i*” indicates the transport direction, we haveGiven that label “*j*” represents the confinement direction, we havewhere is the discrete wavenumber and is the effective mass along the confinement direction. The 2nd term of the right-hand side is the perturbation term raised by the external field. When the semiconductor layer thickness is of the order of nanometers, the contribution of the 2nd term of (11) is quite small. Thus, the effective mass tensor () value of low-dimensional electron systems can be calculated around the subband bottom [15]; the important point is the fact that the effective mass of electrons is larger by the factor of than that estimated assuming a parabolic band.

#### 3. Calculation Results of Effective Mass of Low-Dimensionality Electrons

Assuming a thin Si layer, we calculated effective mass values ( and ) for the (001) surface, where it is assumed that 2-fold and 4-fold -band electrons are confined along the axis. Physical parameters assumed in calculations are summarized in Table 1. Figure 1 shows the effective mass of electrons occupying the ground state as a function of Si layer thickness (), where the nonparabolicity factor is assumed to be 0.5 eV^{−1} and, for comparison, 1st principle calculation results [8, 9] are also shown; the impact of band nonparabolicity on the effective mass of conduction band electrons appears when < 5 nm [8, 9]. The present model successfully reproduces the 1st principle calculation results. Figure 2(a) shows the effective mass of electrons occupying the 1st excited state as a function of Si layer thickness. For comparison, 1st principle calculation results [8, 9] are also shown in Figure 2(a). The present model does not reproduce the 1st principle calculation results, so the conventional value of is not appropriate. We, therefore, varied the value of until the calculated curves fitted the 1st principle calculation results. Figure 2(b) shows the calculation results of the effective mass value of conduction band electrons occupying the 1st excited state, including the nonparabolicity effect, where it is assumed that = 0.1 eV^{−1} for and = 0.05 eV^{−1} for . It is seen that the present model for effective mass successfully reproduces the theoretical simulation results. It is strongly suggested that the reason why the value of for the 1st excited state is smaller than expected stems from the approximation that stripped higher-order expansion terms from (7).