Research note
The effect of image potential on electron transmission and electric current in the direct tunneling regime of ultra-thin MOS structures

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Abstract

The direct tunneling current through ultra-thin gate dielectrics is modeled by calculating the transmission coefficient of an idealized potential barrier that is modified by the image force. A numerical solution to the Schrödinger equation shows that the barrier lowering induced by image-potential affects the tunneling current largely. An analytical expression for the current is obtained within the Wentel–Kramers–Brillouin approximation. The effects of image force on the direct tunneling current are found to increase with the applied voltage across oxide (Vox) and to decrease with the oxide thickness (Tox).

Introduction

It is well known that both image forces and particle tunneling effects give rise to a lower effective barrier height at a metal–semiconductor or metal–insulator interface [1]. Nevertheless it is observed that image-force effects are widely ignored in the context of Fowler–Nordheim (FN) tunneling [2].

The electrical properties of thermally grown ultra-thin SiO2 layers on Si are a critical issue for the characteristics of deep submicron complementary metal oxide semiconductor (MOS) transistors devices. With the thickness of the gate oxide shrinking along with device geometry, for these ultra-thin sub-3-nm gate oxides, direct quantum mechanical tunneling of electrons between the poly-Si and the Si substrate becomes very important [3], [4], [5], [6]. Unlike the leakage mechanism due to FN tunneling [7], [8] and hot carrier injection in devices with thicker oxides, the gate current due to direct tunneling exists even under near-equilibrium conditions in device with ultra-thin oxides, thus having a large impact on power consumption and current drive for logic circuits and causing problems in date retention for memory cells. However, not much is known about the failure induced by direct tunneling for ultra-thin gate oxide. It has been shown that the creation of defects in oxide layer still occurs in the direct tunneling region. A quantitative understanding of degradation induced by direct tunneling will be an important issue in design, optimization of devices in the near future. It is particularly significant to find the effects of image potential on the direct tunneling current. And the image force also helps us to understand the effect of a thin insulating layer which is often used to improve device characteristics.

Section snippets

Fowler–Nordheim tunneling and direct tunneling

The characteristics of tunneling current of thin SiO2 films have been studied theoretically and experimentally [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. For oxide layers thicker than 6 nm and the applied is also in FN region, the leakage current mechanism is explained by FN electron tunneling in MOS structures. The tunneling current density JFN is given by the classical expression [7], [8]:JFN=AEox2exp(−B/Eox)where Eox is the electrical field, the parameters A and B depend

The effect of the image potential on the direct tunneling current

The effect of the image force is to reduce the height of the potential barrier by rounding off the corners and reducing the thickness and, hence, increasing the flow of current between the electrodes. The image potential is a hyperbolic function which, when it is used in obtaining the tunneling current, results in an elliptic integral which can be solved only numerically. Sommerfeld, Bethe and Holm solved the problem analytically by approximating the barrier by a symmetric parabola. This type

The multistep potential approximation

The inclusion of image-force effects require a numerical solution for tunneling current, whether represented in terms of a complete Schrödinger equation. The transmission coefficient was calculated by a numerical solution of the one-dimensional Schrödinger equation assuming an idealized potential barrier. A parabolic E(k) relation with an effective mass m* as parameter is assumed in this article. The barrier was discretized by N partial subbarriers of rectangular shape, which covered the whole

Results and discussion

Electrical transport through thin silicon oxide insulating films is mainly controlled by tunneling. This tunneling current is determined by parameters such as the interface potential barrier height and electron affinity difference between Si and SiO2, especially for silicon oxides thinner than 5 nm [18]. In this article, we improve an analysis by taking into account the barrier height lowering due to the image force. The effects of the image potential on the tunneling current obtained by using

Conclusion

The effects of image potential on tunneling currents are inevitably present in real devices, but may be reduced by putting in the spacer. The effect remains equally important when the current flow is imposed by the external voltage across oxide. Indeed, the value of the barrier is one of the basic parameters that determines whether the device will run in the contact limited or the space limited current regime. However, the precise form of device I(V) characteristics is related to the details of

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