3-D Electrostatic Modeling and Impact of High-κ Control Oxide in Metal Nanocrystal Memory

Although theoretical models of nanocrystal (NC) memories have been investigated by several groups [13], only 1-D electrostatic models were employed, despite the very nature of the 3-D spherical NCs and their 2-D arrayed distribution. In this paper, we establish a physical model based on the 3-D electrostatics for NC memory performance. We demonstrate, by replacing SiO2 with HfO2 as the control oxide in aggressively scaled memories, the continuous control oxide scaling is possible with improved program/erase (P/E) efficiency and retention time owing to the unique 3-D electrostatic effects. The results confirm 3-D electrostatics instead of 1-D should be considered in NC memory modeling. Numerical solution of the 3-D electrostatics in the NC memories is developed to calculate the electrostatic potential, the single-electron charging energy EC, and the 3-D channel-control factor R3D [2, 3], which can not be quantitatively addressed in the previous 1-D models. R3D, less than 1 in general, is a correction factor to the classic flatband voltage shift (AVFB) model in the continuous floating-gate devices due to the partial coverage of NCs over the Si channel. The charges in NCs can only perturb the channel potential in a smaller effective coverage area less than the NC unit cell area, but significantly larger than the NC cross-section area due to 3-D fringing field. Meanwhile, the tunneling calculation at the least-action path is carried out by modified 1-D Wentzel-Kramers-Brillouin (WKB) approximation taking inversion layer quantization into account [4]. The P/E and retention characteristics are determined from time-dependent, self-consistent tunneling current because the potential has to be updated due to the Coulomb blockade effect whenever electrons are in or out of the NCs. Figure 1 illustrates the schematic of the metal NC cell. Good agreements have been shown between the simulated and experimental programming transients of an Au metal NC memory in Fig. 2. Details of the device fabrication were similar to those in [5]. The simulation parameters, initial flatband voltage VFB= 0 V, tunneling oxide thickness Tt,,l = 2 nm, control oxide thickness TC,,l = 27 nm, NC diameter D = 5 nm, NC spacing S = 13 nm, NC density N= 4x 10 cm2 , NC work function 5.1 eV and NC tunneling capture cross-section ACC = 5.3x10-14 cm2 per NC, are fairly close to estimation from various types of physical characterization in the given sample, which validates the accuracy of our formalism. The scaling of the control oxide effective oxide thickness (EOT) is necessary to reduce the memory cell size. Moreover, it increases the coupling ratio, which improves P/E efficiency in the conventional continuous floating gate devices. In the NC memories, however, due to the Coulomb blockade effect, the maximum number of stored charges is self-saturated depending on EC and the bias condition. High coupling ratio by scaling TC,,l may allow more charges stored in NCs at self saturation, but does not guarantee larger AVFB (see Fig. 3), which is influenced by the combined effect of charge density, Tc,,I, and R3D. Even more importantly, if EC does not scale much with Tc,,I, more charges stored in NCs may adversely affect retention characteristics Therefore, the trade-off between Tc,,l and NC memory characteristics is a fundamental problem to consider in the NC memory cell design. High-K dielectrics as the control oxide can provide an effective solution. High-K dielectrics were first introduced to the floating gate devices to suppress the inter-poly leakage [6]. However, the true significance of the high-K control oxide in NC memories lies in the unique 3-D electrostatic nature. As shown in Fig. 4, EC and R3D are relatively insensitive to the control oxide EOT, but strong functions of the dielectric constant. To better understand this, the cross-sections of the 3-D potential contours in the NC unit cell are plotted in Fig. 5. It is obvious that the fringing field through HfO2 to the Si channel is much stronger due to the higher dielectric constant of HfO2. This substantially increases the substrate-NC coupling capacitance and the effective channel coverage area under the influence of the charges stored in NCs. Thus, lower Ec and larger R3D with HfO2 are expected. With the same number of charges stored in NCs, larger R3D benefits the programming efficiency by allowing larger AVFB while lower Ec improves the retention characteristics by maintaining larger band offset between the NC Fermilevel and the Si conduction band. These lead to much improved memory performance in Fig. 6 obtained from our 3-D electrostatic model.