Simulation Techniques for modelling Variability

A range of techniques have been used to model the various effects of intrinsic variability in sub-micron transistors, which range dramatically in terms of the computational complexity. To simulate a device at this level, the self-consistent solution to two equations must be found: these are the Poisson equation, which computes the electrostatic potential between the fixed and mobile charge through the use of boundary conditions, and the transport model equation which models the electron and current density. Three different approaches to solving the equations are outlined here: the simplest is known as Drift-Diffusion simulation (DD), which represents a low order approximation of the Boltzmann transport equation (BTE). A method based on Monte Carlo statistical methods provides an indirect method of solving the BTE, improving on the accuracy offered by drift-diffusion. These two approaches are discussed in greater detail in the following section.

The most complex in terms of computational demand are techniques using quantum- mechanical approaches, most notably Non-Equilibrium Green’s Function (NEGF) modelling. To solve Green’s functions it is necessary to invert very large matrices containing Hamiltonian elements - it is only within the previous year that full 3D-simulations of variability in tran- sistors using NEGF have been undertaken [111]. However, once channel lengths are scaled below ≈ 10nm, quantum effects begin to dominate and the Monte-Carlo approaches lose their validity [138].

Historically, a number of other methods have been used to simulate and model effects of intrinsic variability using a number of different techniques in both 2D- and 3D- domains. Diaz et al proposed an analytical model to estimate the effects of LER on drive current and off-

state leakage in sub-100nm devices. Their model functions by partitioning a device into small unit cells, which each assume a constant gate length [56]. Many trials at combining both DD and Monte-Carlo simulations in hybrid techniques have been developed which use the Monte- Carlo techniques only in regions where it is mandatory to produce accurate results [86].

Drift-Diffusion Simulation

In the simulation of a steady-state NMOS device, the DD approach operates by self- consistently solving the steady-state current continuity equation with Poisson’s equation. To solve these equations it is necessary to calculate the current-density (Jn), which is expressed

as the superposition of the drift component (Jndrift), which is related to the electric field (E),

and the diffusions components (Jndiff), related to the gradient of the electron density (n). The

current-density equations are as follows [184, 138]:

Jndrift = qnµnE (2.20)

Jndiff= qDn

dn

dx (2.21)

where Dnis the diffusion coefficient, q is the electronic charge, and µ is the mobility. The DD

model makes many assumptions to obtain a closed system of equations from the BTE, ignoring non-local effects and assuming thermal equilibrium between the carrier and the lattice. This results in a drawback with the model: its failure to capture non-equilibrium carrier transport effects results in an underestimate of the on-state drain current. The electric field that exists within the active region of a submicron device can be very high, and often undergoes rapid variations over distances which are comparable to the carrier’s mean free path. DD assumes that carriers can make instant responses to changes in the electric field, however in reality they have mass and require a finite time and distance to equilibrate with the field [86, 138].

Despite these shortcomings the DD is still effective when studying the sub-threshold regime of operation in a device due to the weak coupling between current and Poisson’s equa- tion in this region, and as such is suitable for the study of statistical threshold voltage variabil- ity in sub-100nm devices. To improve the validity of such simulations in very small devices it is possible to account for some non-local quantum effects through the use of density-gradient (DG) corrections, which add an extra driving force term to the current density equation [138]:

Jn= Jndrift+ Jndiff+ DG = qnµnE + qDn dn dx + 2qnµn∇  bn ∇2√_n √ n  (2.22)

in which b is a term which expresses the magnitude of the density gradient dependence. This driving force term can be thought of as a “quantum diffusion” current, which has the effect of pushing carriers away from the Si:SiO2 interface. This adds to the model a number

of quantum tunnelling effects, and also has the additional benefit of correctly capturing the effect of RDD on the potential distribution of the device.

Monte Carlo Simulation

In Monte-Carlo Simulation, the Boltzmann transport equation is solved in a stochastic manner, which avoids the limitations that exist in DD simulation. The BTE is virtually impossible to solve directly in all but the most trivial of cases, which results in the popularity of Monte Carlo methods to find solutions. The movement of carries within a device is simulated, which can be considered as a series of free flights and random scattering events at the end of each free flight. The scattering events include interactions between carriers and phonons, fixed impurities and other carriers. The free flight trajectories are calculated using Newtonian physics, with the scattering events adding quantum mechanical calculations. The methods used require a high quality source of random numbers, which are used to determine flight times and scattering events based on appropriate probability distributions.