Multi-Subband Ensemble Monte Carlo simulations of scaled GAA MOSFETs
Introduction
Non-planar MOS transistors with multiple gates represent the most promising solution to the ultimate scaling of CMOS technology [1] because of their superior immunity to short channel effects. Indeed, they are not only a possible future option under scrutiny, but they have already entered mass production [2], [3]. When the size of a device reaches the nanometric scale, quantum effects play an important role; quantum confinement is relevant for reduced lateral size (i. e. in the direction perpendicular to transport) and even more so in non-planar structures where confinement is two-dimensional. An approximately correct electron distribution taking into account quantum confinement can be obtained with corrections to the classical potential, employing algorithms which, however, need proper calibration for each considered structure and crystal orientation (see for example [4], [5]). As a consequence, to properly take into account quantum confinement effects in the cross section of the device it is necessary to solve the Schrödinger Equation (SE). The Multi-Subband Ensemble Monte Carlo (MS-EMC) approach employs the semi-classical Monte Carlo (MC) method to solve the Boltzmann transport equation, coupled with the solution of the SE in the perpendicular direction. The MC method, compared to common transport framework such as Drift–Diffusion, takes directly into account non-equilibrium carrier transport. On the other hand, compared to full quantum approaches, MC allows a simpler implementation of scattering mechanisms and relatively reduced computational requirements. For the same reason, in an effort to improve the classical Monte Carlo and Drift–Diffusion tools without an excessive computational demand, carrier band structure is described analytically employing the (anisotropic) effective mass approximation. Clearly, for severely reduced device dimensions in the transverse direction, this description is not as accurate as the one that could be obtained employing atomistic models such as those based on the Density Functional Theory or Tight Binding approach. However, it has been shown [6], [7] that reliable results can be obtained employing the effective mass model with bulk values of the effective mass for Si nanowires down to a diameter of and even below, if the value of the effective mass is correctly adjusted [7]. For all these reason, the chosen approach can be the optimal middle ground for devices whose transverse dimensions are small enough to make quantum confinement important, yet larger than a few nanometers where atomistic effects become essential.
In this work, we describe an MS-EMC simulator for 3D devices developed by our group, and show the possibilities it offers by studying the scaling properties of ultimate Gate-All-Around (GAA) MOS transistors with ultra-thin Si nanowires. Section 2 is devoted to the description of the simulator, while in the following one (Section 3) we define devices under study and report the obtained results and insights into the device behavior. Finally, conclusions are drawn in Section 4.
Section snippets
Simulator description
To model a non-planar device, such as GAA MOSFETs, we employ a simulator based on the MS-EMC approach. This method has been widely and successfully employed for the simulation of planar semiconductor devices [8], [9], [10], [11], and only recently it has been applied to 3D devices [12], [13]. The simulator is based on the space-mode approach [14], where the SE is solved in several cross sections perpendicular to the transport direction z, for each considered conduction band valley. Each SE
Results
We simulate GAA field-effect transistors based on cylindrical Si nanowires with channel along the direction, with diameter D ranging from to . The choice of the channel orientation is motivated by the better performance predicted for Si nanowires as n-MOSFET with respect to the orientation, given by larger electron mobility [22] and higher ballistic conductance [23]. The coordinate reference system is chosen so that the x and y axes are in the cross section plane and
Conclusion
This paper describes a numerical simulator that solves in a self-consistent way the 3D Poisson equation, the 2D Schrödinger equation, and the 1D Boltzmann transport equation through the Ensemble Monte Carlo method. This software can be used to study the static characteristics of ultra-scaled MOSFET devices, also in the sub-threshold region thanks to the implementation of a variance reduction technique based on variable super-particle weights. It also allows us to inspect the carrier
Acknowledgment
The authors would like to thank the financial support of Spanish Government under project TEC2014-59730-R and EU H2020 program under projects REMINDER (Grant Agreement No. 687931) and WAYTOGO FAST (ECSEL-2014-2-662175).
Luca Donetti received the M.Sc. degree in physics from the University of Parma, Parma, Italy, in 1998 and the Ph.D. degree in physics from the Università degli Studi di Milano, Milano, Italy, in 2002. Since 2005, he has been with the Department of Electronics, Faculty of Sciences, University of Granada, Granada, Spain. His current research interests include the simulation of electron and hole transport properties in nanoscale electronic devices, including SOI and multi-gate devices. In
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2019, Solid-State ElectronicsCitation Excerpt :Finally, the obtained charge distribution is used as an input of the Poisson equation and the whole procedure is repeated in a loop until a self-consistent state is reached. To improve the MC statistics and reduce the statistical noise in the sub-threshold regime, a variance reduction technique based on non-uniform super-particle weight [17] is employed: the weight of super-particles is computed according to their total energy [4]. To improve the performance of the simulator, a high level of parallelization is employed in the MC simulation and in the solution of Poisson and Schrödinger equations.
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Luca Donetti received the M.Sc. degree in physics from the University of Parma, Parma, Italy, in 1998 and the Ph.D. degree in physics from the Università degli Studi di Milano, Milano, Italy, in 2002. Since 2005, he has been with the Department of Electronics, Faculty of Sciences, University of Granada, Granada, Spain. His current research interests include the simulation of electron and hole transport properties in nanoscale electronic devices, including SOI and multi-gate devices. In particular, the focus has been on carrier band-structure effects including simple non-parabolicity models, k·p Hamiltonians up to ab initio calculations, and on quantum effects.