Monte Carlo Study of the Coulomb Interaction in Nanoscale Silicon Devices Nobuyuki Sano  Institute of Applied Physics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan Received August 13, 2010; accepted October 1, 2010; published online January 20, 2011 Three-dimensional Monte Carlo simulations coupled self-consistently with the Poisson equation are carried out under the double-gate metal– oxide–semiconductor field-effect-transistor (MOSFET) structures with various channel lengths. The Coulomb force experienced by an electron inside the device is directly evaluated by performing the Monte Carlo simulations with or without the full Coulomb interaction and the plasmon excitation represented by dynamical potential fluctuations in the source and drain regions by the channel electrons is demonstrated. The drain current and transconductance are greatly degraded below the channel length of 20 nm if the self-consistent potential fluctuations are taken into account and, thus, the Coulomb interaction is indeed a key ingredient for reliable predictions of device properties. # 2011 The Japan Society of Applied Physics 1. Introduction It has been conjectured rather naively that purely ballistic transport would lead to ideal device performance if Si metal–oxide–semiconductor field-effect transistors (MOSFETs) shrink smaller than the mean-free-path of the channel electrons. 1–4) However, it is now recognized that simply scaled Si MOSFETs do not exhibit any performance improvement in deca–nanometer regimes 5,6) and, thus, scattering in the channel still plays an important role and could be inevitable. 7–9) Various scattering processes such as impurity scattering and surface-roughness scattering are proposed to explain such degradation. 10,11) Among others, the Coulomb interaction between the electrons is expected to be of crucial importantance to predict reliable device characteristics. 12) This is because the channel is sandwiched with heavily doped source and drain regions in the distance of tens of nm and the long-range Coulomb interaction is likely to affect the electron transport properties in the channel region. 13,14) Therefore, transport simulations coupled self-consistently with the Poisson equation become mandatory for fully taking account of the dynamical Coulomb interaction. 15) Incorporation of the Coulomb interaction in any particle- based simulations has been a long-standing unsolved problem. It has been tackled with various techniques such as molecular dynamics (MD) simulations, 16,17) the particle– particle–particle–mesh (P 3 M) method, 18,19) and the Monte Carlo (MC) method. 20) The most advanced treatment of the Coulomb interaction in MC simulations so far would be due to Fischetti and Laux, 12,21) by which both the long-range and short-range parts of the Coulomb interaction have been introduced into MC simulations with careful consideration of the dimensionality and the mesh size associated with finiteness of simulated electrons. Unfortunately, their MC simulations have been two-dimensional, yet the potential fluctuations associated with the Coulomb interaction are intrinsically three-dimensional (3D). Hence, we have re- cently extended this approach to 3D MC simulations including the Coulomb interaction as accurately as possi- ble. 22,23) In the present paper, a realistic device structure is introduced into our MC simulator to quantitatively investi- gate the effects of the Coulomb interaction on device char- acteristics under the double-gate (DG) MOSFET structure. The present paper is organized as follows. In x2, the methodology of the MC simulations and the device structures are briefly explained with emphasis on the incorporation of the Coulomb interaction and the degeneracy of electron gas into the MC simulation. The MC simulation results and discussion on device characteristics are given in x3 and conclusions are drawn in x4. 2. Simulation Method and Device Structure 2.1 Coulomb interaction and degeneracy of electron gas In the present study, the conventional MC method coupled self-consistently with the Poisson equation is employed. All relevant scattering processes such as acoustic and optical phonon scattering, impurity scattering, and the short-range electron–electron scattering are included under the frame- work of the nonparabolic band structure of Si. The surface roughness scattering is intentionally ignored in the present simulations in order to look at the intrinsic effects of the Coulomb interaction on electron transport under ultrasmall device structures. As we have reported elsewhere 22,23) the parameter optimization is crucial to incorporate the Coulomb interac- tion accurately into MC simulations. We have performed the MC simulations by artificially turning off all energy- dissipating scattering under thermal equilibrium so that the total energy of the system is strictly conserved. Then, the total energy of the system has been monitored during simulations. This is a very efficient method for parameter optimization to judge the accuracy and the stability of the MC simulations, 18) since energy dissipating scattering always stabilizes the simulations no matter how the simulation parameters are chosen. 24) The time step employed here needs to be small enough to resolve the plasma frequency for stable simulations. 18) We employ the time step of 0.05 to 0.01 fs, which is much smaller than that used in ref. 21 where the time step of 0.2 to 0.4 fs was used. Our optimization procedure is, therefore, much more severe than ref. 21, so that the spatial and temporal resolution of the plasma waves are very high. After careful optimization of simulation parameters such as time step, mesh size, and the size of simulated electrons, our self-consistent MC simulator is now able to reproduce simultaneously both the correct mobility under various impurity concentrations and the collective excitations at plasma frequency. 22) Furthermore, it is necessary to take account of Pauli’s exclusion princi- ple for all short-range scattering to properly simulate the  E-mail address: sano@esys.tsukuba.ac.jp Japanese Journal of Applied Physics 50 (2011) 010108 010108-1 # 2011 The Japan Society of Applied Physics SELECTED TOPICS IN APPLIED PHYSICS Technology Evolution for Silicon Nano-Electronics DOI: 10.1143/JJAP.50.010108