Analytical Modeling of 3D NAND Flash Cell With a Gaussian Doping Profile

Abstract

The incessantly increasing demand for highly dense storage medium in this era of big-data has led to the development of 3D NAND Flash memories. 3D NAND Flash based SSDs have revolutionized edge storage and become an integral part of the data warehouses and the cloud storage systems. The conventional 3D NAND flash memory with greater than hundred stacked word-line (WL) layers suffer from channel tapering and non-uniformity in the threshold voltage of cells in different WL layers which necessitates the use of different programming voltages for different WL layers and a complex error-correction circuitry. A non-uniform vertical (Gaussian) channel doping profile alleviates the threshold voltage non-uniformity and leads to a significant reduction in the complexity of the error-correction circuitry and a simple (uniform) programming scheme for all WLs. Although behavioral models have been proposed to utilize 3D NAND flash memory for circuit and system-level applications, an analytical model is important to understand the intricate details of the device physics and propose design guidelines for efficient cell design. To this end, in this paper, for the first time, we have proposed an analytical model for the characteristic length, surface potential and inner potential of the Macaroni-body 3D NAND flash cell with vertical Gaussian doping profile. A strong agreement between the analytical model and the TCAD simulations for different gate lengths (down to 25 nm), inner and outer radius, channel and oxide thickness of the 3D NAND flash cell validates the efficacy of the developed model.INDEX TERMS 3D NAND flash memory, Macaroni body, Vertical Gaussian doping, Analytical model.

Analytical Modeling of Threshold Voltage and Subthreshold Slope for 3-D NAND Flash Memory With a Non-Uniform Doping Profile

“…The analytical model is derived by simplifying the 3D Poisson's equation (equation ( 1)) to a second-order differential 2)) utilizing Young's parabolic potential approximation [18] (equation ( 3)) and boundary conditions pertaining to the continuity of electrical flux [23].…”

“…1 is symmetrical along the axial direction, results in 𝜓 independent of 𝜃 i.e., 𝜕𝜓(𝑟,𝜃,𝑧) 𝜕𝜃 = 0. Equation (1) will reduce to a 2D Poisson's equation given by [23] as:…”

“…Where 𝐶 1 (𝑧), 𝐶 2 (𝑧) and 𝐶 3 (𝑧) are arbitrary coefficients which can be obtained by using following boundary conditions [23]: a) Electric field lines inside the channel terminate at the silicon body-core filler dielectric interface i.e., the electric field at 𝑟 = 𝑟 1 is zero: where, 𝑡 𝑆𝑖 is the channel thickness, 𝑉 𝑓𝑏 is the flat-band voltage, 𝜓 𝑠 is the surface potential (𝜓 𝑠 = 𝜓(𝑟 = 𝑟 2 , 𝑧)) and 𝐶 𝑜𝑥 is the effective gate oxide capacitance per unit area of cylindrical Macaroni body cell given by [23]:…”

“…Moreover, we define the potential at the channel-core filler interface (𝑟 = 𝑟 1 ) as the inner potential, 𝜓 𝑜 = 𝜓(𝑟 = 𝑟 1 , 𝑧). Now, utilizing above boundary conditions and assumption, equation ( 2) can be expressed as: The inner potential (𝜓 0 at the channel/core-filler interface in figure 1(c)) obtained by solving the differential equation utilizing the principle of superposition where the solution consists of a complementary function and a particular integral [23] as:…”

“…Although a behavioral compact model [16] and an analytical model [17] have been proposed for 3D NAND flash cells with uniform doping profile, an analytical model which provides a physical insight and the necessary design guidelines for 3D NAND Macaroni body flash cell with vertical Gaussian doping profile was elusive. Therefore, we formulated the analytical model for the inner potential and the surface potential of a 3D NAND flash memory with a Gaussian doping profile in the channel region in [23]. However, the stored data is encoded in the threshold voltage of the flash cells in a 3D NAND flash memory and the subthreshold swing (SS) is an important parameter to assess the switching characteristics [24], [25].…”