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.

In the last 3 decades, silicon-on-insulator (SOI) MOSFETs have been attracting attention because of their high short-channel effect immunity [

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.

When an isotropic band structure is assumed, it is conventionally known that its form can be expressed generally as [

When an external field effect is taken account of, we must examine whether the following formulation is theoretically valid or not:

For the two-dimensional electron system, (^{7} V/cm and

The result described above yields an expression for the effective mass tensor (

Assuming a thin Si layer, we calculated effective mass values (^{−1} and, for comparison, 1st principle calculation results [^{−1} for ^{−1} for

Physical parameters assumed in calculations [

Notations | Values | Comments |
---|---|---|

| 0.92 | Longitudinal mass of electrons |

| 0.19 | Transverse mass of electrons |

| 0.49 | Heavy holes |

| 0.16 | Light holes |

| 9.1 × 10^{−31} [kg] | Free electrons |

| 9.7 × 10^{9} [cm^{−3}] | Bulk Si |

| 1.1 [eV] | |

Permittivity of Si ( | 12 | |

Permittivity of SiO_{2} ( | 3.8 | |

Permittivity of vacuum ( | 8.9 × 10^{−14} [F/cm] |

Calculated effective mass values of electrons occupying the ground state as a function of semiconductor layer thickness [^{−1} in order to calculate the nonparabolic band effect.

Calculated effective mass values of electrons occupying the 1st excited state as a function of semiconductor layer thickness [^{−1} in order to calculate the nonparabolic band effect. (b) Effective mass dependence on Si layer thickness. It is assumed that ^{−1} for ^{−1} for

Since the threshold voltage (

On the other hand,

Calculated

^{−1}. The devices shown in [^{−1}. It is also assumed that

Ge-on-insulator (GOI) devices are now attracting attention from the viewpoints of high-speed device applications. However, it is known that Ge demonstrates stronger conduction band nonparabolicity than Si. Therefore, the discussion given here is critical when considering nanoscale GOI device characteristics [

This paper reconsidered the mathematical formulation of the conventional nonparabolic band model. Since the conventional simplified model for band nonparabolicity does not include the perturbation created by the external potential effect, we examined whether such perturbations could be added to the conventional model for convenience. When the perturbation energy is smaller than the unperturbed energy, the insertion of a perturbation term into the conventional expression for the nonparabolic band model is valid; it was confirmed that this approximation is acceptable given a sub-10-nm-thick Si layer surrounded by an SiO_{2} layer. The major discussion focused on the low-dimensional electron system confined by an insulator barrier. For the purpose of verifying this consideration, we addressed the influence of band nonparabolicity on the threshold voltage of Si-based nanoscale SOI FinFETs; calculation results yielded by the proposed model were compared to experimental results and the validity of the model proposed here was confirmed.

In the valence band, it is well known that the nonparabolic band energy dispersion relation of holes can be approximately expressed as [

Our solution is to rewrite (

The author declares that there are no competing interests.

A part of this study was partially conducted by MEXT-Supported Program for the Strategic Research Foundation at Private Universities, “creation of 3d nano-micro structures and its applications to biomimetics and medicine,” 2015–2019.