The Monte Carlo method for electron transport is a semiclassical Monte Carlo (MC) approach of modeling semiconductor transport. Assuming the carrier motion consists of free flights interrupted by scattering mechanisms, a computer is utilized to simulate the trajectories of particles as they move across the device under the influence of an electric field using classical mechanics. The scattering events and the duration of particle flight is determined through the use of random numbers.
Background
Boltzmann transport equation The Boltzmann transport equation model has been the main tool used in the analysis of transport in semiconductors. The BTE equation is given by:
∂ f ∂ t + 1 ℏ ∇ k E ( k ) ∇ r f + q F ( r ) ℏ ∇ k f = [ ∂ f ∂ t ] c o l l i s i o n {\displaystyle {\frac {\partial f}{\partial t}}+{\frac {1}{\hbar }}\nabla _{k}E(k)\nabla _{r}f+{\frac {qF(r)}{\hbar }}\nabla _{k}f=\left[{\frac {\partial f}{\partial t}}\right]_{\mathrm {collision} }}
v = 1 ℏ ∇ k E ( k ) {\displaystyle v={\frac {1}{\hbar }}\nabla _{k}E(k)}
The distribution function, f, is a dimensionless function which is used to extract all observable of interest and gives a full depiction of electron distribution in both real and k-space. Further, it physically represents the probability of particle occupation of energy k at position r and time t. In addition, due to being a seven-dimensional integro-differential equation (six dimensions in the phase space and one in time) the solution to the BTE is cumbersome and can be solved in closed analytical form under very special restrictions. Numerically, solution to the BTE is employed using either a deterministic method or a stochastic method. Deterministic method solution is based on a grid-based numerical method such as the spherical harmonics approach, whereas the Monte Carlo is the stochastic approach used to solve the BTE.
Monte Carlo method The semiclassical Monte Carlo method is a statistical method used to yield exact solution to the Boltzmann transport equation which includes complex band structure and scattering processes. This approach is semiclassical for the reason that scattering mechanisms are treated quantum mechanically using the Fermi's Golden Rule, whereas the transport between scattering events is treated using the classical particle notion. The Monte Carlo model in essence tracks the particle trajectory at each free flight and chooses a corresponding scattering mechanism stochastically. Two of the great advantages of semiclassical Monte Carlo are its capability to provide accurate quantum mechanical treatment of various distinct scattering mechanisms within the scattering terms, and the absence of assumption about the form of carrier distribution in energy or k-space. The semiclassical equation describing the motion of an electron is
d r d t = 1 ℏ ∇ k E ( k ) {\displaystyle {\frac {dr}{dt}}={\frac {1}{\hbar }}\nabla _{k}E(k)}
d k d t = q F ( r ) ℏ {\displaystyle {\frac {dk}{dt}}={\frac {qF(r)}{\hbar }}}
where F is the electric field, E(k) is the energy dispersion relation, and k is the momentum wave vector. To solve the above equation, one needs strong knowledge of the band structure (E(k)). The E(k) relation describes how the particle moves inside the device, in addition to depicting useful information necessary for transport such as the density of states (DOS) and the particle velocity. A Full-band E(K) relation can be obtained using the semi-empirical pseudopotential method.
Hydrodynamic and drift diffusion method Both drift diffusion (DD) and the hydrodynamic (HD) models can be derived from the moments of the Boltzmann transport equation (BTE) using simplified approximation valid for long channel devices. The DD scheme is the most classical approach and usually solves the Poisson equation and the continuity equations for carriers considering the drift and diffusion components. In this approach, the charge transit time is assumed to be very large in comparison to the energy relaxation time. On the other hand, the HD method solves the DD scheme with the energy balance equations obtained from the moments of BTE. Thus, one may capture and calculate physical details such as carrier heating and the velocity overshoot effect. Needless to say, an accurate discretization method is required in HD simulation, since the governing equations are strongly coupled and one has to deal with larger number of variables compared to the DD scheme.
Comparison of semiclassical models
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