A 2D threshold-voltage model for small MOSFET with quantum-mechanical effects
Introduction
As complementary metal–oxide–semiconductor transistor (CMOS) integrated circuit (IC) technology enters the deep-submicron level, it is required that oxide thickness (Tox) is decreased to several nm’s, while substrate doping concentration (Nsub) is increased to >5 × 1017 cm−3 [1]. The two factors produce a very high vertical electric field at the Si/SiO2 interface, leading to more serious quantization effects on the carriers in the inversion channel of MOS field-effect transistor (MOSFET), which obviously affect the design and electrical properties of devices. These effects become increasingly large as oxide thickness decreases further, resulting in large error in the simulated electrical properties of devices based on the classical method. For example, for Nsub = 5 × 1017 cm−3, the simulated threshold-voltage of MOSFET could have a 0.1 V error. Therefore, it is necessary to develop an accurate threshold-voltage model including these effects.
Since Stern proposed a self-consistent solution of the Poisson’s and Schrödinger’s equations to analyze the quantum effects [2], [3], many relevant researches have been reported. Based on energy-subband analysis and modifying the classical 1D threshold-voltage model, Van Dort etc. proposed a semiempirical model which exhibits good agreement with experimental data [4]. Ma proposed concepts of quantum effective state density to study the influence of quantum-mechanical effects (QME) on carrier concentration and surface potential [5]. All these works concentrate on QME in the direction normal to the Si/SiO2 interface, and thus the resulting 1D models are not suitable for short-channel MOSFET.
In this work, an accurate 1D threshold-voltage model modified by quantum effects is firstly proposed from self-consistent numerical solution and experimental data. Then, it is extended to 2D situation, and a 2D threshold-voltage model including both short-channel effect (SCE) and QME is developed for short-channel MOSFET.
Section snippets
Self-consistent solution of Poisson’s and Schrödinger’s equations
When gate oxide is down to about 3 nm and substrate doping is up to around 5 × 1017 cm−3, vertical electric field becomes large enough to cause significant quantization of carrier energy and a redistribution of carriers at the Si/SiO2 interface. Energy-band diagram in the inversion region is illustrated in Fig. 1. For simplicity, a uniformly-doped device is assumed, with same doping in the bulk and channel in the following discussion.
The QM approach of a 1D potential-well system requires solving
QM-modified threshold-voltage model for long-channel MOSFET
Impact of quantum effects on threshold-voltage is as follows:
- (1)
The classical threshold condition becomes not suitable for quantization case.
- (2)
QME lead to a different distribution of electron concentration. n(z) vanishes at the Si/SiO2 interface and the highest concentration occurs in the Si body. This makes the average distance between carriers and the interface 〈z〉 increased by an amount Δz relative to the classical distribution, as shown in Fig. 3. An additional potential drop occurs in the
Quantum-modified threshold-voltage model for small MOSFET
In 2004, Intel announced to produce MOSFET with channel length L = 35 nm and Tox = 1.2 nm using 65 nm CMOS technology. Apparently, the classical theory of MOSFET is not suitable for such short channel because the QME and SCE must be considered to describe the threshold-voltage behavior of sub-micron and deep-submicron MOSFET’s.
For including SCE in the threshold-voltage model, the effects of transverse electric field in the channel direction (y-direction) have to be taken into account. So, it is
Comparison of threshold-voltage models
For understanding the influences of SCE and QME on threshold-voltage, it is necessary to compare threshold-voltages from different models, i.e. classical threshold-voltages for both long-channel and short-channel MOSFET’s without considering the quantum effects, and also the quantum-modified threshold-voltages for long-channel and short-channel MOSFET’s. They are denoted as VTH0, VTH1, VTH2, VTH3, respectively. VTH2 and VTH3 are given in Eqs. (13), (17), respectively, while VTH0 and VTH1 have
Summary
A 2D quantum-modified threshold-voltage model is developed for small MOSFET by using self-consistent numerical solution and considering the 2D electric field distribution in the channel region. The model exhibits good agreement with experimental data, and thus suitable choices of device parameters can be made. Furthermore, the model can be used for deep-submicron MOSFET with high-k gate-dielectric. Therefore, it has high potential in the design of small MOSFET and simulation of ULSI circuits.
Acknowledgement
This work is financially supported by the National Natural Science Foundation of China (NSFC, Grant No. 60376019) and Open Foundation of State Key Laboratory of Advanced Technology for Materials Synthesis and Processing (Project No. WUT2006M02).
References (10)
- et al.
Nano CMOS device
(2004) Phys Rev B
(1972)- et al.
Phys Rev
(1967) IEEE Trans Electr Dev
(1992)- Ma YT, Li ZJ, Liu LT, Yu Z. In: Proceedings of the international conference on microelectronics, vol. 1; 1999. p....