Non-Planar MOSFET Modeling with Analytical Approach

Non-planar structures have been identified as promising structure for next device generation in the nanoelectronic era. However, the continuous device dimension scaling into nano regime eventually requires more sophisticated model due to the limitation of the existing models. A model for non-planar MOSFET structure was elaborated in this paper, especially for device with pillar structure, using analytical approach. The concern of channel shape and structure were discussed as well. The result shows the shift in subthreshold characteristic in the channel with recessed channel model. The charge sharing is suspected as one of the key parameter in the shift of performance in the recessed region.


Introduction
The rapid development of MOSFET have shown a tremendous progress in downscaling the dimension and structure as well into nanoscale.The continuous scaling of the device dimension has now reached tens of nanometer size especially for the channel length of transistors.With the smaller device dimension, it is getting impossible to maintain the conventional bulk structure legacy, as shown by International Technology Roadmap for Semiconductor (ITRS) [1].Several future devices with non-conventional structure have been proposed to overcome the conventional MOSFET structure limitations, as example in [2]- [8].The non-planar MOSFET is expected to expand the Moore's law.Several structures could shift the downsizing off the lithography and obtain self-aligned multi gate which is hard to produce with conventional one.
On the other hand, several fabrication methods produces various channel shapes, Some methods result in straight channel between source and drain, while some others create such a bending channel, a combination of vertical and recessed horizontal direction of current.For example, several geometries may be found in non-planar double gate MOSFET.Figure 1(a) reveals the recessed channel with L-shaped geometry due to the presence of corner that diverts the direction of current from drain to source.The second geometry, shown in Figure 1(b), offers the possibility of direct flux.This body-tied geometry have its channel connected to the substrate potential.The floating body channel (Figure 1(c)) reveals simple double gate structure but with its channel potential isolated electrically from the substrate.
Several models that simulate the physics of the vertical devices have been published recently.However, many authors focused on the ideal non-doped double-gate/surround gate MOSFET structure [9]- [11].Others modeled the highly doped MOSFET with the help of regional model [12], [13] or depletion charge [14] as well as with conformal mapping [15].However, the analytical model for non-ideal junction structure, as well as recessed channel, has not been well articulated in previous publications.In the conventional MOSFET, the recessed gate is known for its ability to prevent short channel effect, but in the vertical geometry, the recessed channel is somewhat different with that of conventional.The simulation of recessed gate in vertical MOSFET geometry has been presented in [16], but it is mainly concentrated for vertical surround gate, and no analytical model was offered.Therefore, in order to explain the behaviour of channel with different shape and potential profile, it is important to derive the appropriate model.In addition, the continuous scaling of device requires applicable model in nanoscale.This paper elaborates the analytical model of non-planar MOSFET which employ the recessed channel geometry in the bottom part.The corner effect is elaborated extensively, with the help of approach that has been provided for conventional model [17].In addition, the preceeding reports on this model have also been reported in [18].This extended report provides more evaluation on the short channel effects which are crucial in the nano regime.

Analytical Model
The structure of recessed channel in non-planar MOSFET can be seen in Figure 2.This typical structure is an ideal form of the fabricated devices.The straight channel potential model is derived from Poisson equation using generic approach as noted in several references, e.g.[10], [19]- [22]: Moreover, the presence of corner region in the bottom is arguably difficult to solve using two-dimensional Poisson equation in Cartesian system.Therefore, we divide the channel into two regions: first is the straight and latter is the corner region.We simplify the corner region as a quarter circle, an approach adopted from Zhang et al [17] for grooved gate in conventional MOSFET.However, Zhang's model used trapezium-shape approach, while our model employs quarter-circle approach which is more realistic.
The uniformed depletion width is calculated using the following formula: ( is the total depletion region area as in [23], as simplification of depletion width towards all area mainly due to the gate influence.The potential of channel for each region is following the two dimensional second-order parabolic approach of graded channel approximation (GCA) that originally proposed by Young [24].However, for corner area, the parabolic approximation is adopted to the cylindrical coordinate system as was used in [17].The potential for both cartesian and polar coordinate systems are: The notation "st" stands for "straight", to differentiate with "cr" for "corner".In solving the differential Poisson equation using general parabolic approach, several boundary conditions are set, which are applied to both regions (for polar coordinate, x and y should be replaced with and r, respectively): The electric field in the silicon-oxide interface is according to Gauss' law [23]: ( (i) The potential at depletion layer is equal to the substrate potential, for body-tied channel to the V sub : , (ii) The electric field in the depletion layer is: 0 All parameters are put and later the potential equation is rewritten as: Similarly for the corner area, with inner radius , depletion depth and capacitance of cylindrical tube / ln 1 ⁄ as derived in  ISSN: 1693-6930 TELKOMNIKA Vol. 12, No. 4, December 2014: 779 -786 782 [25], by borrowing quarter circle approach of grooved channel, solving all boundary conditions for corner area's potential result in [17]: where: Eq. 9 shows that the potential equation throughout the corner are dependent of , which represent the length of recessive part of the channel.By substituting potential equation , of Eq. 8 back into Poisson's equation, a differential equation of potential in straight channel is obtained: Similarly for corner area, the surface potential of Eq. 9 is substituted into Poisson's equation for polar coordinate [17]: which result in [17] ′′ It is noteworthy that Poisson equation of both regions as expressed in Eqs. 10 and 12 can be rearranged in the generic second-order differential equation formula [26] which has the common form of: ′′ , for corner The respective boundary conditions for both regions can be defined as: (i) ψ 0,0 0 Threshold voltage V T is determined when the value of minimum surface potential is twice the Fermi potential , 2 .Thus, the location of the minimum potential along the surface in the channel obeys the expression: By solving the boundary conditions and calculating the location of minimum potential in each regions, the threshold voltage can be determined from the minima of both regions.

Results and Analysis
The structure of straight channel with no corner effect was simulated as well as the recessed structure.The surface potential of recessed and non-recessed channel are shown in Figure 3 as a function of channel length for a fixed recessed length, L rec =10 nm.For the bodytied structure, the body of channel outside the depletion region is assumed to have the potential of substrate, .Meanwhile, Figure 4 shows the surface potential with the recessed channel is limited to around 15% of the total channel length L. It is notable that the minimum surface potential is located in the straight region for L rec < 40% L. Figure 5 reveals the extracted threshold voltage for both structures with L rec = 15% L, V ds =0.1 V, N A = 10 18 cm -3 , t ox =2 nm.The threshold voltage shows the tendency of decreasing in shorter channel, as is expected due to short channel effect.In addition, the straight channel obtains lower V T than the recessed channel.In any application with the usage of low V DD which is commonly found in low-power nano IC, lower threshold voltage is required for a better switching operation, which can maintain the lower drain voltage as well.Furthermore, lower V T may produce higher current drive for operations with high power.
Physically, the curved channel structure lacked the gate control in the corner region.The similar phenomenon is also found in grooved channel in planar MOSFET, as in [27], [28].The lack of gate control produces decreased potential in the corner area compared to the straight channel.It also prevents the quick conversion into inversion in the channel beneath the oxide layer, with the charge slowly respond to the gate voltage.As a result, higher threshold voltage is needed in the recessed channel.In the presence of higher drain voltage, the control of charge inversion is affected.The value of DIBL is shown at Figure 6, calculated for at V ds =0.1 and 1.0 V.The curved channel suffers from higher DIBL in short channel compared to the straight one.The curved channel has a higher potential barrier at the drain end, but the increase of drain potential suppresses this barrier height more than in the straight counterpart.With the benefit of lower DIBL at short channel, and more acceptable threshold value in nanoscale device operation, the option of straight channel is more preferable than that suppressed channel counterpart.Therefore, the ORI-based vertical MOSFET offers advantages in the short channel regime.

Conclusion
The non-planar MOSFET structure with recessed channel has been modeled analytically using parabolic approach.Two distict region representing different channel shape were applied for the ease of model.The simulation based on the developed model shows the shift of threshold voltage due to the presence of the recessed part in the corner.In addition, the DIBL reveals the tendency of increasing in the lower channel length, as a manifestation of the short channel effect.While the straight channel offers lower threshold voltage, the higher DIBL is found for recessed channel.The result reveals the implication of the usage of pillar for nonplanar structure, which requires more careful design in the future for threshold-sensitive application .

Figure 1 .
Figure 1.The variation of channel geometry in non-planar DG MOSFET: (a) recessed, L-shape channel, (b) body-tied channel, and (c) floating-body channel

TELKOMNIKAFigure 2 .
Figure 2. The structure of recessed channel in vertical geometry (a), and its approach using two-region solution (b) MOSFET Modeling with Analytical Approach (Munawar A. Riyadi) 783

Figure 5 .Figure 6 .
Figure 5.The threshold voltage due to short channel effect