Monte Carlo Simulation of Quantum Transport in Semiconductors Using Wigner Paths

Abstract

Charge transport in mesoscopic semiconductor systems must be analyzed in terms of a quantum theory since nowadays typical dimensions of the physical structures are comparable with the electron coherence length. Theoretical approaches based on fully quantum mechanical grounds have been developed in the last decade with the purpose of analyzing the quantum electron-phonon interaction in electron transport. The Wigner function (WF) formalism is particularly suitable for the analysis of mesoscopic structures owing to its phase-space formulation that allows a natural treatment of space dependent problems with given boundary conditions. The Hamiltonian describing the system is [1] $$ H = - \frac{{\hbar ^2 }} {{2m}}\nabla ^2 + \sum\limits_q {b_q^\dag } b_q \hbar w_q + \sum\limits_q {i\hbar F\left( q \right)} \left( {b_q e^{iqr} - b_q^\dag e^{ - iqr} } \right) + V(r) + eE.r $$ where the terms in the RHS describe, respectively: free electron evolution, free evolution of the phonon system, electron-phonon interaction, structure potential, constant uniform accelerating field. Due to the linearity of the Liouville equation, these various contributions can be independently developed in the equation of motion for the generalized WF including phonon variables [2], leading to an equation of the form: % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbuLwBLnhiov2DGi1BTfMBaeHb % d9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaadeWaaq % aadaqbaaGcbaacdiGaa8Nzaiaa-DhadaqadaqaaGWaaiaa+jhacaGF % SaGaa4hCaiaa+XcadaGadaqaaGqaciab95gaUnaaBaaaleaacaGFXb % aabeaaaOGaay5Eaiaaw2haaiabcYcaSmaacmaabaGaemOBa42aa0ba % aSqaaiaa+fhaaeaaiiGacqaFYaIOaaaakiaawUhacaGL9baacqGGSa % alcqWG0baDaiaawIcacaGLPaaacqGH9aqptuuDJXwAK1uy0HwmaeXb % fv3ySLgzG0uy0Hgip5wzaGGbaiqb7ftigzaaiaWaaeWaaeaacqqF0b % aDdaWgaaWcbaGaeeimaadabeaaaOGaayjkaiaawMcaaiabdAgaMjaa % -DhacaWFRaWaa8qaaeaafaqabeGabaaabaGae0hDaqhabaGae0hDaq % 3aaSbaaSqaaiabbcdaWaqabaaaaOGaa4hzaiab9rha0jab8jdiIcWc % beqab0Gaey4kIipakmaadmaabaacgiGafOxfXBLbaGaadaqadaqaai % abdsha0jab8jdiIcGaayjkaiaawMcaaiabgUcaRiqb79q8qzaaiaWa % aeWaaeaacqWG0baDcqaFYaIOaiaawIcacaGLPaaaaiaawUfacaGLDb % aacqWGMbGzcaWF3bGaa8hlaaaa!7ED3! \[ fw\left( {r,p,\left\{ {n_q } \right\},\left\{ {n_q^\prime } \right\},t} \right) = \tilde \mathcal{F}\left( {t_{\text{0}} } \right)fw + \int {\begin{array}{*{20}c} t \\ {t_{\text{0}} } \\ \end{array} dt\prime } \left[ {\tilde \mathcal{V}\left( {t\prime } \right) + \tilde \mathcal{P}\left( {t\prime } \right)} \right]fw, \] where F is the operator describing the ballistic free evolution of the WF, while V(t) and P(t) are the operators accounting respectively for potential and phonon scattering at time t. Eq. (1) is formally analogue to the Chambers transport equation for the Boltzmann distribution, so even the numerical solution technique and the physical interpretation may be quite similar to a semiclassical approach. This equation is in fact iteratively solved by Monte Carlo sampling, and the concept of Wigner paths is introduced [1],[3]; they are formed by ballistic flights during which the constant field E acts, interrupted by scattering events due to other fields or phonons, in analogy to the semiclassical case. The WF is supposed to be known inside the simulated device at a given initial time, and at device boundary at every time. The value of the WF at a given phase-space point at a time t can be constructed as the sum of the contributions due to a very large number of Wigner paths that terminate at that point and start at points (and times) in which the WF is known. These contributions are weighted by the proper quantum phase a nd by a factor due to the scattering mechanisms acting along each path.