extrapolation of lifetime. Further, failure is usually a combination of factors – like electric field (gate oxide degradation in Si MOS), current density (electromigration in back-end wiring), mechanical stressing (contact plug delamination), or complex interactions of these effects (gate sinking driven by field and mechanical stress). In these cases, the simple temperature extrapolation becomes irrelevant to the actual lifetime.

In these more complex cases, accurate modeling is vital to predicting the lifetime. This is where simulation can play a role. A reliability simulator should be able to simulate the conditions of the stress test, accurately reproduce results, and then use those physical insights to predict normal operation lifetime. This chapter describes an approach used to build such a simulator and has two operating examples.

16.2FLOORS Overview

16.2.1 Finite Element Approach

A finite element discretization method (FEM) conveniently fulfills the need for fully coupled, multiphysical modeling that is essential in reliability studies. Thus, FLOORS is based upon this approach. Electronic operation of semiconductor devices may depend on temperature and mechanical strain. Thus, the partial differential equations governing electronic operation, thermal transport, and elastic deformation will have various degrees of coupling within different simulations. Interaction between the three physical domains (electrical, thermal, and mechanical) is simplified by the use of the same basic functions in the discretization of each differential equation via the FEM.

The partial differential equations commonly used within the domains are given below.

The first two equations (16.1) and (16.2), comprise the three-coupled partial differential equations commonly used for simulation of electronic operation in semiconductor devices. First is the Poisson equation, 16.1, where C is electrostatic

potential, n is the density of electrons, p is the hole density, NA and ND are acceptor and donor impurity densities, respectively, q is charge, and e is permittivity. The electron and hole current continuity equations are given in 16.2, where Jn and Jp are the electron and hole current densities, respectively, and r and g are recombination and generation terms. The thermal domain is characterized by the heat conduction equation given by 16.3, where T is temperature, c is the specific heat capacity, K is thermal conductivity, and Q is the heat generation term. The mechanical equilibrium equation, 16.4, governs elastic deformation, where stress s is related to strain e by the constitutive relation s ¼ Dðe e0Þ. D, the stiffness matrix, is a function of Young’s modulus and Poisson’s ratio. Discretization of the equations within each domain has conventionally been accomplished with different methods.

Device simulation was made possible by the seminal work by Scharfetter and Gummel [1]. Finite difference or finite volume methods have been extensively used for discretization of the electrical partial differential equations for more than 40 years, e.g., (16.1) and (16.2) [2]. The Scharfetter–Gummel (SG) discretization method is applied to the continuity equations of Eq. 16.2 as a way to avoid instabilities otherwise found in a standard difference scheme [1]. Within this technique, the current density is formulated in terms of drift and diffusion components as shown in Eq. 16.5:

where m is electron mobility and D is electron diffusivity. A similar equation arises for hole current density. The SG method solves this equation one dimensionally assuming constant electric field. Thus, the current is defined along the grid edges with appropriate weighting factors. The SG method cannot rigorously be used with the FEM since it does not have an expression that continuously defines current in space. Some disadvantages of the SG method are that current is only defined on the edge of an element and thus evaluation of the total current over the element is not exact, plus current flow becomes dependent on the grid [2]. However, this method is very stable and reliable and remains the approach for many device simulators including PICES-II, PADRE, Sentaurus SDevice, and Silvaco Atlas.

Though using finite element methods to discretize the Poisson and continuity equations is less common, this approach is viable [3–5]. For formulation in a finite element-based discretization scheme, the continuity equations are written in terms of the electron and hole quasi-Fermi levels (fn,p) as in Eq. 16.6:

as compared to the drift-diffusion formula that is implemented with a finite volume SG scheme. Recent work has shown that simulation results obtained by the finite volume SG method or quasi-Fermi FEM are comparable and that the FEM approach is preferable for situations when current flow is not in the direction of the grid as in ionizing radiation in single event upsets [6]. Another advantage for

The PartialDiffEq class contains one or more terms that have various derived pieces. Every portion of the term is parsed into a binary expression tree, with the usual numerical operators, simple functions, and logical expressions supported.

For assembly of the equations, both the value and the derivative of the expressions need to be computed for the Newton’s method linearization. Each expression knows how to differentiate itself, and the chain rule is used liberally to construct the derivative expressions. For small simulations, the overhead of computing the derivatives can become a substantial fraction of the computation time. For most simulations, however, this computation is negligible.

The equations are then assembled using finite element techniques, as mentioned previously. The assembly is performed over 128 elements at a time. This allows most operations to work on 128 long vectors of values, which makes efficient use of modern pipelined architectures. To further save computation time, results are cached. Expressions that are evaluated multiple times are computed just once for each batch of 128 elements. This allows an automated efficiency in reusing results that requires no user intervention. Initial results on reasonably complicated problems (point-defect-mediated diffusion in two dimensions) show that this approach is comparable in CPU requirements as a hand-coded approach.

The great advantage of this is that physics can be adapted and improved quickly. In a research environment in which the dominant physics is not quite well understood, it allows the simulator to be adapted quickly and efficiently. Examples of adding complicated physics are contained in later sections.

16.2.3 Calibration Example: GaN HEMT

Device simulations using the Florida object-oriented reliability simulator (FLOORS) were calibrated by comparing results with those from commercially available software, Synopsys Sentaurus, under identical device structure and electrical conditions. The GaN/AlGaN HEMT structure used in both simulators consisted of an undoped 1.0-mm SiC substrate, an undoped 0.20-mm AlN layer, a 1.775-mm GaN layer with an acceptor doping level of 6.5e16 cm 3, and an undoped 0.025-mm Al0.26Ga0.74N layer (Fig. 16.2). The structure was 4.0 mm wide. A 0.025- mm t-gate was contacted at the center of the surface of the AlGaN layer, and source and drain were contacted to the AlGaN at the left and rightmost regions of the layer. An insulating oxide layer was created on top of the AlGaN layer that also surrounded the contacts. An interface charge of 1.17e13 was placed between the AlGaN and GaN layers to reflect the polarization of the crystal due to the material offset at the interface.

A gate voltage, VG, was applied in 1-V steps from 0 to 4 V, while the drain voltage, VD, was swept from 0 to 10 V at each gate voltage step. The initial device temperature of the simulations was 300 K. A wafer backside temperature of 300 K was also incorporated in the simulations. Contact resistance for both source and drain was set at 0.5 Ω.

Fig. 16.2 Cross-sectional diagram of the device structure used in FLOORS simulations

Development of a robust simulation model relied on incorporation of accurate expressions for mobility, electric field, thermal transport, and trap activity within the device. Bearing in mind that these simulations were focused on a p-type HEMT, with electrons flowing in the channel within the GaN layer, the equations for electrons and the physical parameters of GaN will be specifically emphasized here due to their overwhelming impact within the device.

Low and high field electron mobility expressions for GaN that identically matched those in the Sentaurus simulations were used in FLOORS:

with the values for the saturation velocity, Vsat, and b shown in Table 16.1. The expression for the electric field, E, was the first derivative of the electrostatic potential C.

The electron mobility was used in the thermal transport equation for GaN (eII. a.3), which accounts for changes in the device temperature profile as a result of carrier movement and lattice thermodynamics [8]. The heat capacity, c, and thermal conductivity, k, of GaN are given in Table 16.1. The final term of Eq. 16.3 is the heat generation term, Q, which represents self-heating within the device due to power dissipation and other carrier/lattice interactions. From the several models for Q that have been proposed [9–11], FLOORS currently uses Eq. 16.7 for the heat

Fig. 16.3 ID vs. VD for FLOORS (blue) and Sentaurus (green) for VG ¼ 0 to 4 V and VD ¼ 0–10 V

generation [12], which is simplified in the FLOORS to include only the n type due to negligible hole concentration for the simulated HEMT structure. The definition of Jn, the electron current density, is given in Eq. 16.6. Other parameters for the model are in Table 16.1.

Employing these thermoelectric models within FLOORS resulted in the IV curves shown in Fig. 16.3. The saturation voltages, VDsat, for the FLOORS and Sentaurus curves matched well for all VG values. The saturation voltages, IDsat, for both simulators were also almost identical for each gate voltage, with the largest discrepancies (~5%) occurring at the lowest VG values. A noticeable difference between the FLOORS and Sentaurus curves occurred at high VD values, where ID for FLOORS was lower than Sentaurus by approximately 0.10 A/mm (11%) at VD ¼ 10 V for VG ¼ 0 V. As VG decreased, the difference lessened, such that at VG ¼ 4 V, the curves were identical. The discrepancy at high VD can likely be