Uncertainty quantification for charge transport in GNRs through particle Galerkin methods for the semiclassical Boltzmann equation

TL;DR

This work develops an efficient uncertainty quantification framework for charge transport in graphene nanoribbons by extending particle-based simulations of the semiclassical Boltzmann equation with stochastic Galerkin projections. The approach combines a DSMC-like collision scheme with a Galerkin reconstruction to preserve positivity and key physical properties while handling uncertain band gaps and applied fields via a gPC basis. Numerical results demonstrate robustness to width and field uncertainties, quantify stochastic errors, and show favorable comparisons to Monte Carlo sampling, including substantial speed-ups for low-cost scenarios. The method provides a practically relevant tool for predicting GNR performance under fabrication and operating variabilities, with potential extensions to adaptive and higher-fidelity random parameter models.

Abstract

In this article, we investigate some issues related to the quantification of uncertainties associated with the electrical properties of graphene nanoribbons. The approach is suited to understand the effects of missing information linked to the difficulty of fixing some material parameters, such as the band gap, and the strength of the applied electric field. In particular, we focus on the extension of particle Galerkin methods for kinetic equations in the case of the semiclassical Boltzmann equation for charge transport in graphene nanoribbons with uncertainties. To this end, we develop an efficient particle scheme which allows us to parallelize the computation and then, after a suitable generalization of the scheme to the case of random inputs, we present a Galerkin reformulation of the particle dynamics, obtained by means of a generalized Polynomial Chaos approach, which allows the reconstruction of the kinetic distribution. As a consequence, the proposed particle-based scheme preserves the physical properties and the positivity of the distribution function also in the presence of a complex scattering in the transport equation of electrons. The impact of the uncertainty of the band gap and applied field on the electrical current is analysed.

Figures (6)

• Figure 1: Schematic representation of a graphene nanoribbon. Note the irregular edges.
• Figure 2: Test 1: Comparison between Parallel (red circles) and Non Parallel (solid black line) algorithms of the time evolution of energy $\mathcal{E}(t)$ (left column) and the longitudinal component of the electron velocity $V_x(t)$ (right column). The number of particles is fixed to $N=10^6$, the time step is $\Delta t=0.0025\,\mathrm{ps}$, the Fermi energy is $\varepsilon_F=0.4\,\mathrm{eV}$. The electric field $E_x$ and the GNR width $W$ are varied according to titles of the subfigures.
• Figure 3: Test 2: Time evolution of the expectations w.r.t. $\mathbf{z}$ of the energy $\mathcal{E}(t,\mathbf{z})$ (left column) and the longitudinal component of the electron velocity $V_x(t,\mathbf{z})$ (right column). In all the simulations, we consider uncertainties in the GNR width $W(\mathbf{z})=(6.5+\mathbf{z}) \, \mathrm{nm}$, with $\mathbf{z}\sim\mathcal{U}([0,1])$. The number of particles is $N=10^6$, the time step is $\Delta t = 0.0025 \, \mathrm{ps}$. The Fermi energy $\varepsilon_F$ and the electric field $E_x$ are varied according to the titles of the subfigures.
• Figure 4: Test 2: Time evolution of the expectations w.r.t. $\mathbf{z}$ of the energy $\mathcal{E}(t,\mathbf{z})$ (left column) and the longitudinal component of the electron velocity $V_x(t,\mathbf{z})$ (right column). In all the simulations, we consider uncertainties in the applied electric field $E_x(\mathbf{z})=(0.4+0.2\mathbf{z}) \, \mathrm{V} / \mu \mathrm{m}$, with $\mathbf{z}\sim\mathcal{U}([0,1])$. The number of particles is $N=10^6$, the time step is $\Delta t = 0.0025 \, \mathrm{ps}$. The Fermi energy $\varepsilon_F$ and the GNR width $W$ are varied according to the titles of the subfigures.
• Figure 5: Test 3: Comparison of the sG Error in the evaluation of the energy $\mathcal{E}(t,\mathbf{z})$ at fixed times $t=0.1,\,0.5,\,1\,\mathrm{ps}$, for increasing $M$. Left panel: uncertain GNR width $W(\mathbf{z}) = ( 6.5 + \mathbf{z} ) \, \textrm{nm}$, $\mathbf{z}\sim\mathcal{U}([0,1])$ and $E_x=0.5\,\mathrm{V}/\mu\mathrm{m}$. Right panel: uncertain electric field $E_x(\mathbf{z}) = (0.4+0.2\mathbf{z}) \, \mathrm{V}/\mu\mathrm{m}$, $\mathbf{z}\sim\mathcal{U}([0,1])$, and $W=7\, \textrm{nm}$. In all the simulations $N=10^4$, the time step is $\Delta t=0.0025\,\mathrm{ps}$, and the Fermi level is $\varepsilon_F=0.4\,\mathrm{eV}$. Reference solution computed with $M^{\textrm{ref}}=30$.

Theorems & Definitions (2)

• Remark 1
• Remark 2