On the electrostatic behavior of floating nanoconductors

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Abstract

In this work, the electrostatic behaviour of nanometer-sized metallic floating conductors embedded in a dielectric medium has been investigated. First of all, we present a semi-analytical approach based on Green functions and show its excellent quantitative agreement with well known finite element methods concerning electrostatic computations. Then, we compare floating potential values obtained by numerical simulation with an approach based on an equivalent capacitor circuit. We show that the latter is inappropriate for sub-100 nm wide floating electrodes when the parallel plates approximation (PPA) is used to estimate the coupling capacitance values because of lateral coupling effects. Nevertheless, we finally show that these coupling capacitances can be easily extracted to predict the electrostatic potential of metallic nano-dots with the same accuracy than with a direct numerical computation. We finally propose a figure of merit to estimate the validity of the PPA.

These results are useful to predict the floating potential of future generations of non-volatile-memories using metallic dots. Moreover they can be used to give a first estimation of the floating potential in a self consistent Poisson–Schrödinger resolution when considering semiconductor nanocrystals.

Introduction

In the field of non-volatile memory devices (NVM), replacing the traditional floating gate by silicon nanocrystals is a promising way to push the downscaling of these devices toward sub-100 nm technological nodes [1], [3]. Using this technique, high density silicon nanocrystals, up to 1012 cm−2, are required to achieve a sufficient programming window and reduce electrical fluctuation from one device to the other [1], [4], [5]. Recent papers have shown that using metallic dots can easily provide high-density 2D-arrays and maintain compatibility with standard CMOS technology [6], [8].

Modelling the electrical characteristics of these devices, such as programming (resp. erasing) and retention characteristics, requires the knowledge of the floating potential value (Vf) of each storage dot. In the case of semiconductor nanocrystals, due to the pronounced quantum confinement effects, this potential can be numerically obtained by a self-consistent resolution of Poisson and Schrödinger equations [9], [10], [11]. On the contrary, in the case of metallic islands, an electrostatic approach based on coupling capacitances can be used to predict Vf. Nevertheless, many works still apply this latter approach, derived form flash memories models [12], even in the case of semiconductor nanocrystals ([20], [21], [22], [23], [24]). Moreover, the coupling capacitances used in these cases correspond to the parallel plates approximation (PPA), shrinked to the dot diameter. This assumption can be easily understood when the floating conductor is heavily doped and sufficiently wide to screen the top electrode (i.e. the control gate) from the other (i.e. the substrate surface), like in the case of flash memories. However, considering ultra-scaled memory devices employing metallic islands or even semiconducting nanocrystals this assumption should be discussed. Indeed, in these devices individual electrical behavior of storage nodes, such as single electron effects, becomes quantitatively more significant than their collective behavior [2]. In this context, it is interesting to study the individual properties of these storage nodes. In particular, the PPA, which may be suitable when there is enough dots above the active area to assume a collective behavior, should be discussed when considering a single dot.

In the first section of this paper, we briefly expose the basis of the equivalent capacitor circuit and the PPA. Then, we present a semi-analytical method, based on Green functions, suitable to study the electrostatic properties of metallic floating nanoconductors. We compare this method with finite element method (FEM) results and show their excellent agreement when simulating memory stacks containing a single island with different shape. In the last section, the coupling capacitances of these memory stacks are extracted and compared to the PPA. We demonstrate that the large mismatch observed between both methods can be explained by the fact that PPA is inadequate for sub-100 nm wide floating conductors. To conclude, we propose a figure of merit suitable to estimate the validity of the PPA.

Section snippets

Equivalent capacitor circuit

Let us consider the simple case of a floating metallic island embedded in a dielectric medium between two metallic plates. The metallic nature of the island implies the presence of a large amount of electronic states at Fermi level and thus allows to treat it as if it was a perfect conductor [7]. Following this assumption, an equivalent capacitor circuit can be defined as depicted in Fig. 1 to predict the floating potential of the island. In the case of NVM devices the same approach is used to

Simulated devices

The general description of the simulated devices is shown in Fig. 2. A dielectric of permitivity εox=ε0·εrSiO2(εrSiO2=3.9) lies between two metallic planes supposed to be perfect conductors. Let V1 and V2 be the electrostatic potential of these conductors, named S1 and S2. A metallic dot S3, representing the floating conductor, is embedded in the dielectric medium and supposed to be perfect conductor too. This conductor gets the floating potential Vf, constant on its surface. Let us note Q0 the

Application to floating nanoconductors

In this section, we present the extraction of the coupling capacitances by the two presented methods for the three configurations previously presented (see Fig. 3). These results are confronted to the corresponding parallel plates approximation and a discussion on the limits of this approximation is proposed.

Conclusions and perspectives

In this work the electrostatic properties of metallic floating nanostructures embedded in a surrounding dielectric have been studied. First of all, a semi-analytical method based on Green functions has been presented and successfully confronted to FEM simulations to predict the floating potential of nanometer-sized metallic dots. Then, we compared these results with predictions given by an equivalent capacitor model using shrinked parallel plate as coupling capacitors. A large mismatch has been

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