Hierarchical Device Simulation

The ongoing advance of CMOS based microelectronics is mainly due to the continuous reduction of the device feature size which is expected to decrease for at least another decade [1.1–1.4]. State of the art are gate lengths of 0.13µm, volume production of devices with about 0.10µm gate length is now beginning, and technologies with much shorter gate lengths are under development [1.5]. Additional performance improvements beyond device scaling are obtained by using improved device structures, such as SOI (e.g. [1.6]) or FinFETs (e.g. [1.7]). The introduction of the SiGe technology has opened up new possibilities previously only available in expensive III-V technologies like band-gap engineering and enhancement of carrier mobility by strain [1.8–1.11]. By using strained Si layers pseudomorphically grown on relaxed SiGe layers in the channel region of CMOS devices the performance of MOSFETs has considerably been improved [1.12–1.15]. The performance of Si BJTs has been enhanced by fabricating the base with strained SiGe pseudomorphically grown on the Si bulk [1.16–1.18].

Here, the classical theory of a kinetic gas is applied to the electron and hole ensembles in semiconductors with two quantum mechanical extensions. The particle kinetics are based on a position-dependent band structure calculated with the nonlocal empirical pseudopotential method [2.1–2.3] and scattering rates determined by Fermi’s Golden Rule [2.4, 2.5]. In this theoretical framework the particle motion consists of a series of scattering events and accelerations by external forces, which is described by the semiclassical Boltzmann transport equation (BTE) [2.4–2.10].

The MC method is a stochastic method for the solution of integrals [3.1–3.4]. By formal integration the BTE is transformed into an integral equation, which can be solved with the MC method [3.5–3.12].

In a semiconductor the charge carriers are scattered by various mechanisms [4.1–4.6]. They are scattered by phonons, which are the quasiparticles of the lattice vibrations (Sec. 4.1), alloy disorder in composites (Sec. 4.2), ionized dopants (Sec. 4.3), impact ionization (Sec. 4.4), or microscopically roughness of interfaces between different materials (Sec. 4.5).

The complicated dependence of the energy on the wave vector makes it impossible to capture all details of the band structure by analytical approximations and the full details of the band structure (full-band) are included in the MC model based on a numerical representation of the band structure [5.1]. The basic properties and symmetries of the band structure of RSi are discussed in the first section of this chapter. The more general case of strained SiGe follows in the next section. The grid and the interpolation method for the energy in the KS are developed in the third section. Based on this grid efficient methods for the calculation of the density of states are discussed in the fourth section and a formulation of the mass tensor consistent with an unstructured tetrahedral grid is given in the fifth section. Methods for the motion of particles in the KS are presented in the sixth section and CPU efficient methods for the selection of the final state are given in the seventh section.

In the case of a device simulation the BTE has to be solved self-consistently with the Poisson equation for the electric field [6.1]. To this end the RS is discretized with a tensor-product grid as described in the first section of this chapter. The material parameters, like the germanium concentration, doping, etc, are defined on this grid together with the boundary conditions. The germanium-dependent band edges are given in the next section. The discrete Poisson equation is presented next and in the fourth section the self-consistent solution of the BTE and Poisson is discussed. The extension to a nonlinear Poisson equation based on the zero-current approximation for one carrier type is given in the following section. Nonself-consistent MC simulations, where the electric field is calculated with a momentum-based method, are introduced in the sixth section. A method for the enhancement of rare events (e.g. impact ionization) is discussed in the next section. In the following two sections methods for the evaluation of terminal currents and inclusion of contact resistances are presented. Finally, the normalization of physical quantities is discussed in the last section of this chapter.

The MC model is very CPU intensive and simpler but more CPU efficient models based on balance equations have been developed. The most widely used momentum-based models are the drift-diffusion (DD) and the hydrodynamic (HD) models [7.1–7.13]. Both models are derived by applying different degrees of approximation to balance equations of the type (2.39). The DD model is the simplest momentum-based model and consists of balance equations for the particle and current densities. Thus, only the first two moments of the distribution function are calculated instead of the full distribution function and a large fraction of the information contained in the distribution funtion is lost. On the other hand, the dimentionality of the problem is reduced by the integration of the k-space and the CPU efficiency is improved by orders of magnitude. In the case of the HD model the first four moments are considered including the particle density, current density, particle gas temperature, and the energy current density. This already enables the simulation of nonlocal effects,like the velocity overshoot, which have a strong impact on the device behavior of modern deep sub-micron devices.

The relative stochastic error of a quantity evaluated by stationary MC simulation is proportional to its variance, which can be evaluated with the corresponding autocorrelation function (cf. Sec. 3.4).

In this chapter the transport models are applied to the simulation of three different types of devices to assess the accuracy of the transport models either by comparison to each other or experimental data. In addition, the transport in the different device types is analyzed with the appropriate simulation model. The first set of devices, unipolar 1D N⁺NN⁺ and P⁺PP⁺ structures, is mainly of academic interest and is frequently investigated because of its relative simplicity (e.g. [9.1–9.6]), which reduces the CPU time and makes it easier to compare the MC model to the momentum-based models. The second group of devices consists of CMOS devices, which are the standard devices of today’s integrated circuits. With the introduction of the SiGe technology many new types of CMOS devices have appeared [9.7, 9.8]. The most important types are MOSFETs with strained Si channels [9.9–9.13]), which are investigated together with standard Si MOSFETs in the second part of this chapter. In addition, PMOSFETs with a strained SiGe channel, which have promising potential for low noise amplifiers, are discussed [9.14–9.18]. The third group of investigated devices consists of SiGe HBTs, which exhibit a superior RF and noise performance compared to standard Si BJTs [9.19–9.22].