Many-particle Hamiltonian for open systems with full Coulomb interaction: Application to classical and quantum time-dependent simulations of nanoscale electron devices

Abstract

A many-particle Hamiltonian for a set of particles with Coulomb interaction inside an open system is described without any perturbative or mean-field approximation. The boundary conditions of the Hamiltonian on the borders of the open system ͓in the real three-dimensional ͑3D͒ space representation͔ are discussed in detail to include the Coulomb interaction between particles inside and outside of the open system. The manyparticle Hamiltonian provides the same electrostatic description obtained from the image-charge method, but it has the fundamental advantage that it can be directly implemented into realistic ͑classical or quantum͒ electron device simulators via a 3D Poisson solver. Classically, the solution of this many-particle Hamiltonian is obtained via a coupled system of Newton-type equations with a different electric field for each particle. The quantum-mechanical solution of this many-particle Hamiltonian is achieved using the quantum ͑Bohm͒ trajectory algorithm ͓X. Oriols, Phys. Rev. Lett. 98, 066803 ͑2007͔͒. The computational viability of the manyparticle algorithms to build powerful nanoscale device simulators is explicitly demonstrated for a ͑classical͒ double-gate field-effect transistor and a ͑quantum͒ resonant tunneling diode. The numerical results are compared with those computed from time-dependent mean-field algorithms showing important quantitative differences.

Many-particle Monte Carlo Approach to Electron Transport

“…it describes the time-evolution of the electron distribution function (i.e. the Boltzmann distribution) in a single-electron phase-space (Albareda, López, Cartoixà, Suñé & Oriols, 2010;Albareda, Saura, Oriols & Suñé, 2010;Albareda, Suñé & Oriols, 2009;Boltzmann, 1872). Moreover, in addition to the above problems, due to the computational burden associated to the microscopic description of electron transport, the simulation of the Coulomb correlations between the electrons in the leads and those in the active region of an electron device is not always possible and the use of small simulation boxes is a mandatory requirement in modern nanoscale simulators (see Fig.…”

“…In this chapter, we are interested in revisiting the MC computation of an ensemble of Coulomb interacting particles in an open system (an electron device) without any of the approximations mentioned in the previous paragraphs. With this goal, in section 2 , we will develop an exact many-particle Hamiltonian for Coulomb interacting electrons in open systems in terms of the solutions of multiple Poisson equations (Albareda, Suñé & Oriols, 2009). To our knowledge, the type of development of the many-particle Hamiltonian proposed here has not been previously considered in the literature because, up to now, it was impossible to handle the computational burden associated with a direct solution of a many-particle Hamiltonian.…”

“…Section 4 , will be devoted to present a semi-classical solution of the time-dependent many-particle Hamiltonian provided with the previous boundary conditions. This solution constitutes a generalization of the semi-classical single-particle Boltzmann distribution to many-particle systems (Albareda, López, Cartoixà, Suñé & Oriols, 2010;Albareda, Suñé & Oriols, 2009). In section 5, our many-particle MC will be used to evaluate the importance of accounting for strongly-correlate phenomena when predicting discrete doping induced effects in the channel of a quantum wire double-gate field-effect transistor (Albareda, Saura, Oriols & Suñé, 2010).…”

“…Now, it can be shown that the error of the time-dependent mean-field approximation, (Albareda, Suñé & Oriols, 2009):…”

“…Here is precisely where the MC technique comes into play by providing a sequence of random numbers with such particular distribution probabilities. The study of electron transport through the BTE constitutes, however, just a single-particle approach to the electron transport problem (Albareda, Suñé & Oriols, 2009;Di Ventra, 2008), only accurate enough to describe electron dynamics in large structures containing a large number of carriers. As electron device sizes shrink into the nanometer scale, device structures are characterized by simultaneously holding a small number of electrons in a few nanometers.…”