Wigner Monte Carlo Approach to Quantum Transport in Nanodevices P. Dollfus, D. Querlioz, J. Saint-Martin, V.-N. Do, A. Bournel Institut d'Electronique Fondamentale, Univ. Paris-Sud, CNRS Bat. 220, 91405 Orsay, France Email : philippe.dollfus@ief.u-psud.fr Abstract—The Wigner Monte Carlo approach is shown to provide an efficient way to study quantum transport in the presence of scattering and to connect semi-classical to quantum transport. The study of resonant tunneling diodes highlights the physics of the impact of scattering on resonant tunneling, and on electron decoherence and localization. The simulation of nano- MOSFET evidences a mixed regime, where both quantum transport and scattering play a significant role. Keywords: Tunneling, Wigner distributions, Monte Carlo methods, Quantum theory, MOSFETs, Green function, Resonant Tunneling Diodes I. INTRODUCTION The particle-based Monte Carlo technique has been, and is still, widely used to study the physics of transport in semiconductor devices within the semi-classical Boltzmann approximation. The extension of this method to quantum transport in nanodevices has been proposed through the Wigner function formalism. Though the Non Equilibrium Green’s Functions (NEGF) formalism is now often used to simulate devices operating in the quantum regime [1-6], it remains uneasy to include realistic scattering models because of theoretical and computational difficulties. In particular, NEGF modeling cannot readily use all the work done to model scatterings in semi-classical transport situations. The Wigner quasi-distribution function (WF) can provide an original alternative to NEGF. It is defined in the phase-space as the center-of-mass Fourier transform of the density matrix of the carriers, and is most interesting because it shares many similarities with a distribution function, although it is not positive-definite [7]. In fact it tends to the Boltzmann’s equation distribution function in the semi-classical limit. Besides, scattering effects can be incorporated into a WF calculation with an approach similar to that used for the Boltzmann’s equation. WF thus provides a rich formalism to study the transition from semi-classical to quantum transport in nanoscaled electron devices. Our Monte Carlo device simulator has been recently extended to include the WF evolution while maintaining full compatibility between semi-classical and quantum transport descriptions. This paper describes the main features of this approach and presents some important results regarding scattering effects in typical devices operating in quantum regime. II. PRESENTATION OF THE MODEL The motion equation of the Wigner function f w (Wigner Transport Equation, WTE) reads * w w w w f f k Qf Cf t m x ∂ ∂ + = + ∂ ∂ = (1) where w Qf is the quantum evolution term resulting from the non-local effect of the potential ( ) U x , defined as ( ) ( ) ( ) 1 , ' , ' , ' 2 w w w Qf xk dk V xk f xk k π = + ∫ = (2) where the Wigner potential is given by w V ( ) ( ) ' ' , ' sin ' 2 2 w x x V xk dx kx U x U x ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = + − − ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ∫ (3) The term w Cf in Eq. (1) encodes the effect of electron scattering (typically by phonons, impurities or oxide roughness) on the Wigner function. In a one-particle approach, and within an instantaneous scattering approach (that neglects quantum collision effects like collision broadening and intra- collisional field effect [8]), the collision Hamiltonian leads to the standard collision term of the Boltzmann equation of semi- classical transport [9] ( ) ( ) ( ) ( ) ( ) , , , , , w w i i w i Cf xk dk f xk S k k dk f xkS kk ⎡ ′ ′ ′ = ⎣ ⎤ ′ ′ − ⎦ ∑ ∫ ∫ (4) where i refers to the type of scattering mechanism and i S is the associated scattering rate. To solve the WTE we use the particle Monte-Carlo interpretation inspired by [10] and described in [11, 12]. The Wigner function is seen as a sum of “pseudo-particles” localized in both x and k space and weighted by a parameter called affinity. The affinity contains the quantum information on the particles and can take negative values. Such pseudo- particles have no direct physical meaning, but constitute a very ingenious way to solve the WTE: their x and k coordinates This work was partially supported by the European Community, through NoE NANOSIL and IP PULLNANO, and by the French ANR, through project MODERN.