1. Introduction

Electrical performance of on-chip interconnects becomes increasingly important. As the circuit density steadily increases, the operating frequencies and clock rates get higher, and the die size has been actually increasing, the effects of on-chip interconnects become a limiting factor to the overall circuit performance. The device performance improves with each new chip generation. On the other hand, the higher level of integration brings more and longer wiring interconnections onto a chip. The on-chip wiring delays become even more significant portions of the total chip delays than before. Moreover, many other interconnect effects such as losses, dispersions, and crosstalks may degrade the performance of circuits. Accurate simulations of the on-chip interconnects are required in order to predict the circuit performance confidently.

The metal-insulator-semiconductor (MIS) interconnects, being one of the most elementary components in the modern integrated circuits, have been of fundamental interest. Slow-wave propagation in MIS and Schottky-contact interconnects has been both experimentally observed and theoretically explained from different points of view [1]-[8]. The slow-wave properties of such interconnects can be employed to reduce the sizes and cost of distributed elements to implement delay lines, variable phase shifters, voltage-tunable filters, etc. Moreover, the energy dissipation in both semiconductor layer and conductor may also have significant impacts on the performance of MIS interconnects. In order to understand various mechanisms such as slow-wave effect, semiconductor losses, conductor losses, crosstalks, dispersions, or reflections at discontinuities, it is necessary to accurately simulate the MIS interconnect structures. In the following sections, the physical models and numerical schemes used in the previous work for studying MIS interconnects are reviewed.

2. Physical Models

The physical models employed for the performance study of the MIS interconnects include:

• Analytical or empirical lumped circuit models
• Parallel-plate waveguide models
• Planar multi-layered multi-conductor transmission line models
• Combined electromagnetic and device simulation models

1. Analytical or empirical lumped circuit models

Based on the assumption that the conditions for slow-wave propagation mode are satisfied, an analytical model of the Schottky contact coplanar line was presented in [9]. The circuit parameters were determined using the formulation from the previous work of the coplanar waveguide. However, this analytical model may not be applicable to either the skin-effect mode or the dielectric quasi-TEM mode.

In [10], an analytical lumped circuit model for MIS interconnects was obtained using the energy and power conservation laws. The circuit parameters were calculated using analytical or empirical formulations derived from simplified structures. An MIS interconnect system was decomposed into two isolated parallel line systems: a coplanar line system without substrate and a quasi-microstrip line system with the substrate serving as the return path. Note that the validation of isolated structure decomposition may hold only for relative simple structures.

The analytical lumped circuit models provide fast calculation and first-hand insight to the performance of MIS interconnects. However, they are, in general, restrictedly applicable to only some simplified situations. Nevertheless, the analytical model can be employed to initialize more powerful numerical algorithms.

2. Parallel-plate waveguide models

Many early studies of MIS interconnects utilized the parallel-plate waveguide models. In [3], the effects of the substrate conductivity and frequency were investigated using so-called "wide-strip" limit, i.e., parallel-plate waveguide model. Existence of three fundamental modes: "dielectric quasi-TEM mode", "skin-effect mode", and "slow-wave mode" is demonstrated, and the condition for the occurrence of each mode was illustrated on the resistivity (of the semiconductor)-frequency plane. The physical mechanism of these three propagation modes was explained thoroughly. An equivalent circuit for each of these modes was presented.

An analysis of Schottky contact microstrip lines was given in [11] using the parallel-plate waveguide approach. The metallic losses were included in this analysis. Formulas for the propagation constant and characteristic impedance were derived and an equivalent circuit is presented.

The parallel-plate waveguide models are the simplest electromagnetic simulation models for MIS interconnect structures and can be readily solved using the classical mode-matching method [12-14]. In fact, the parallel-plate waveguide models usually result in a one-dimensional (1-D) electromagnetic analysis. They are capable of providing the first-order approximation and revealing the physical mechanism of the electromagnetic wave propagation along the MIS interconnects, although they neglect the fringing effects of the finite narrow conductor lines.

The primary information obtained from the simulation includes:

• Electric field and magnetic field

Note that the electric and magnetic fields are directly attained if a differential equation approach is used. When an integral equation approach is used, the current and charge distributions on the signal line are calculated and then the electric and magnetic fields are computed by convolution integrals.

The following quantities can be derived from the primary information:

• Line voltage and line current
• Propagation constant, attenuation factor and slow-wave factor
• Characteristic impedance
• Current distribution in semiconductor layer
• Equivalent circuit

1. Planar multi-layered multi-conductor transmission line models

In order to accurately simulate the MIS interconnects and confidently predict their performance, the real MIS interconnect configuration should be used in the analysis. The general MIS interconnect configuration consists of dielectric (insulator) layers, semiconductor layers, and multi-conductor lines, which is best described by a planar multi-layered multi-conductor transmission line model. Most of recent research work on the MIS interconnects is based on this model.

The planar multi-layered multi-conductor transmission line models usually receive full-wave analysis treatments. The commonly used numerical techniques for the full-wave analysis include the spectral domain analysis (SDA) method [13], [15-18], the method of lines [16], the transmission line matrix (TLM) method [19], the finite-difference time-domain (FDTD) method [20], and the finite element method [21-23].

The planar multi-layered multi-conductor transmission line models make use of the planar property of the MIS interconnect configuration and generally require a two-dimensional (2-D) electromagnetic analysis. These models are capable of providing the information on the effects of conductor losses, semiconductor losses, dispersion, crosstalks, and slow-wave propagation on the performance of MIS interconnects. The semiconductor effects are simply accounted for in these models by virtue of the conductivity and dielectric constant of semiconductor material.

The primary information obtained from the simulation includes:

• Electric field and magnetic field

Note that the electric and magnetic fields are directly attained if a differential equation approach is used. When an integral equation approach is used, the current and charge distributions on signal lines are calculated and then the electric and magnetic fields are computed by convolution integrals.

The following quantities can be derived from the primary information:

• Line voltages and line currents
• Scattering parameters, i.e., S-parameters of multi-port circuit network
• Characteristic impedance matrix
• Impedance matrix or admittance matrix
• Current distribution in semiconductor layer
• Equivalent multi-port circuit network

Note that the S-parameters are reduced to the set of propagation constant, attenuation factor and slow-wave factor for a one-signal-line system.

1. Combined electromagnetic and device simulation models

When an electromagnetic wave propagates along an MIS structure, the screening effect of the carriers in the semiconductor prohibits the field from penetrating deep into the semiconductor, in addition to that caused by attenuation effect arising from energy dissipation. In order to describe the behavior of semiconductor as solid state plasma, a transport-based analysis [24, 25] is required. In other words, the coupled electromagnetic and device simulation models need be solved to include the interaction mechanism between the electromagnetic field and the carriers in the semiconductor.

In [26], the propagation property of the fundamental mode in a biased parallel-plate MIS waveguide was investigated using a transport-based analysis. A formulation incorporating Maxwell's equations and the equations of motion of the carriers was solved using the 1-D finite difference scheme.

The combined electromagnetic and device simulation models allow the treatments of carrier accumulation and depletion as well as the screening effect of carriers, in addition to providing the information on the propagation properties of electromagnetic waves along the MIS structures. The MIS waveguide models in [26] may need to be generalized to 2-D models such as multi-layered multi-conductor transmission line structures for the general configuration of MIS interconnects.

A three-dimensional (3-D) combined electromagnetic and device simulation model was presented in [27] and [28] to study the performance of microwave devices. Maxwell's equations in conjunction with a 3-D hydrodynamic model are solved using the finite-difference time-domain (FDTD) method. This approach includes the interactions of the conducting carriers with the electromagnetic wave. However, this approach has been found to be computationally intensive. The simulation was performed on a massively parallel machine. The sinusoidal excitation has been used in the FDTD simulation and thus the Fourier transform is avoided. However, it loses the most attractive feature of a time-domain analysis that one could get the data in a wide range of frequencies with only a single simulation in the time domain.

The primary information obtained from the simulation includes:

• Electric field and magnetic field
• Electron and/or hole concentrations

Note that a differential equation approach may always be required since an integral equation is unlikely to be available for the coupled electromagnetic and device simulation problem.

The following quantities can be derived from the primary information:

• Line voltages and line currents
• Scattering parameters, i.e., S-parameters of multi-port circuit network
• Characteristic impedance matrix
• Impedance matrix or admittance matrix
• Current distribution in semiconductor layer
• Equivalent multi-port circuit network

Note that the S-parameters are reduced to the set of propagation constant, attenuation factor and slow-wave factor for a one-signal-line system.

1. Numerical Algorithms for Electromagnetic Analysis

The commonly used approaches for the electromagnetic analysis can be roughly classified into three categories:

• Quasi-TEM analysis
• Frequency-domain full-wave analysis
• Time-domain full-wave analysis

1. Quasi-TEM Analysis

Due to the inhomogeneous dielectric geometry and energy dissipation, rigorously speaking, an MIS interconnect can not support a TEM wave. However, when the transverse dimensions are much smaller than a wavelength, a quasi-TEM assumption may be applicable [29, 30]. It is reported that the validity of the quasi-TEM assumption holds for switching speeds below 7ps [30].

The quasi-TEM analysis is based on the potential theory. The original vector field problem is reduced to a scalar potential problem. Hence, the problem becomes much easier to be solved and the computational efforts are much less than a full-wave analysis.

2. Frequency-Domain Full-Wave Analysis

The frequency-domain full-wave analysis solves one of the three equivalent sets of equations [13] [15-26]:

• Time-harmonic Maxwell's equations
• Helmholtz wave equations
• Integral equations derived from the wave equations using Green's functions

The first two sets of equations lead to differential equation schemes such as the finite difference method (FDM) or the finite element method (FEM), while the last set of equations results in integral equation schemes such as the boundary element method (BEM), the method of moments (MoM), or the spectral domain analysis (SDA) method.

In a differential equation approach, the differential equation formulation associated with the electromagnetic field problem under study is solved directly using numerical algorithms. Hence, a sparse matrix equation is yielded, but a mesh volume much larger than the interested region is usually required such that an absorbing boundary condition can be applied.

Alternatively, one can convert the differential equation formulation into an integral equation formulation using Green's function. Based on the integral equation formulation, an integral equation approach can be established. In the integral equation approach, the computation domain is limited in the exactly interested region and a dense but small matrix equation is usually resulted.

• Mode Matching Method: In the mode-matching method, the MIS interconnect configuration is identified as two or more regions, each of which belongs to a separable coordinate system [13]. The basic idea in the mode-matching method is to expand the unknown fields in the individual regions in terms of their respective normal modes and then the continuity conditions on the interfaces of regions are imposed. The mode-matching method is computationally efficient method. However, it may be applicable only to some relative regular structures such as parallel-plate waveguides, since the general complex configuration may not be amenable to be decomposed into regular regions.

• Spectral Domain Analysis (SDA): The spectral-domain analysis method is an integral equation approach. The integral equation is derived from the time harmonic Maxwell's equations or Helmholtz wave equations using Green's function. For the MIS interconnects, Green's function is better formulated in the spectral domain by virtue of the planar structure [13] [15-18]. Since the discretization is required only on the surface of conductors, the spectral domain analysis results in a very small dense matrix equation and thus leads to a pretty fast scheme. The major drawback of the SDA is that it is not easy to be generalized since Green's function may not be necessary to be available for a general configuration and inhomogeneous material distribution.

• Method of Lines: The method of lines converts a partial differential equation into an ordinary differential equation by partially discretizing the given problem. Then the ordinary differential equation can be solved analytically. The spectral domain approach is not suitable for analyzing the propagation characteristics of the gradually inhomogeneously doped MIS coplanar waveguide. The method of lines can be used in this case [16].

• Finite Difference Method (FDM): The finite difference method approximates the differential operators by the finite differences. The finite difference method is easy to implement and applicable to the general configuration. However, the finite difference method has difficulty to handle curved boundaries. As a differential equation approach, the FDM may need a large mesh volume to implement the absorbing boundary condition for unbounded problems. The FDM provides a straightforward and efficient discretization scheme for the bounded problems such as the MIS coplanar waveguides [26].

• Frequency-Domain Transmission Line Matrix (FDTLM) Method: Like the FDM, the FDTLM method discretizes the solution domain in a mesh volume [31]. The electromagnetic field is considered at each node of the mesh. The interaction of electromagnetic field among the nodes is governed by the so-called scattering matrix at each node. In contrast to the time-domain transmission line matrix (TDTLM) method [32, 33], in which an impulse excitation leads to a time-iterative procedure of scattering at each node and interconnection with its neighboring nodes, the FDTLM algorithm works entirely in the frequency domain. That is, the FDTLM network is considered to be in a steady-state where scattering and interconnection occur simultaneously. The advantage of this approach is that time synchronism (required by the TDTLM) is not needed and that a graded mesh layout with large grading ratio can be chosen. This is particular useful in the MIS interconnect analysis because of the very thin inhomogeneous substrate-layers that are in close proximity to larger structural details [19]. The FDTLM has the same limitations of the FDM.

• Finite Element Method (FEM): In the finite element analysis, the solution domain is discretized and represented as a patchwork of elements [34]. By virtue of the interpolation functions, each of the elements is mapped into a basic standard element. The unknown fields are locally expressed in terms of the interpolation functions over each individual element. A set of algebraic equations is then obtained by applying the variational or Galerkin procedure. The FEM is the most general approach and applicable to the most general configuration. The salient merit of the FEM is its capability of handling the curved boundaries and arbitrary inhomogeneous material distribution. As a differential equation approach, one need pay attention to the implementation of absorbing boundary conditions for unbounded problems. Another potential problem is the possible existence of the spurious modes. The FEM has been applied to study the performance of the MIS interconnects [21-23].

• Boundary Element Method: The FEM method can be applied to the boundary integral equations, which results in the boundary element method [35,36]. The boundary element method has both the merits of the FEM and an integral equation approach, i.e., the capability of modeling the arbitrarily curved boundaries and small mesh volume. However, the boundary integral equations may not be necessary to be available for the inhomogeneous material distribution.

The frequency-domain full-wave analysis solves the problem at each interesting frequency and thus can easily incorporate various frequency-dependent interactions such as dispersions, skin effects, reflections at discontinuities, losses, etc. For transient analysis, the frequency-domain full-wave analysis may be less efficient than the time-domain full-wave analysis.

3. Time-Domain Full-Wave Analysis

In general, the time-domain full-wave analysis starts on the original Maxwell's equations [27,28]. The most popular schemes for the time-domain full-wave analysis are the finite-difference time-domain (FDTD) method and the transmission line matrix (TLM) method.

• Finite Difference Time Domain (FDTD) Method: The FDTD approach directly solves Maxwell's time-dependent curl equations. It is based upon volumetric sampling of the unknown near-field distribution within and surrounding the structure of interest, and over a period of time [27, 28, 37, 38]. The sampling in space is at sub-wavelength resolution to properly sample, in the Nyquist sense, the highest near-field spatial variations. Typically, 10 to 20 samples per wavelength are required. The sampling in time is selected to ensure numerical stability of the scheme. The FDTD is a marching-in-time procedure. The primary merit of the FDTD method is that it is a natural method for analyzing the transient responses and possible to perform wide-band analyses with a single computation process. As a differential equation approach, one need pay attention to the implementation of absorbing boundary conditions for unbounded problems. In general, the FDTD is computationally expensive and has difficulty to handle frequency-dependent interactions such as material dispersion and metal skin effects. Another thing needed to pay attention is that the frequency-domain parameter extraction using the Fourier transform is quite sensitive to the tails of time-domain responses.

• Time Domain Transmission Line Matrix (TDTLM) Method: The transmission-line matrix (TLM) method is originally a time-domain method [32, 33]. Recently, it has been extended into the frequency domain [31]. The time-domain TLM (TDTLM) simulates the electromagnetic wave propagation in the time domain by discretizing the solution domain into a mesh volume like the FDTD method. It therefore has a number of similarities with the FDTD method (see below). The method models the spatial electromagnetic field in terms of a distributed transmission line network. Electric and magnetic fields are made equivalent to voltage and current on the network. The numerical simulation starts by exciting the transmission line matrix (network) at specific points using voltage or current pulses. Propagation of the pulses in the matrix (network) is then evaluated at discrete time intervals. Time synchronism is required so that all pulses reach nodes at the same time. The major merit of the TDTLM method is the simplicity of implementation and the capability of analyzing the transient responses. Basically, all the limitations of the FDTD method are also applicable to the TDTLM method.

Recently, it has been proven that both the FDTD method and the TLM method can be obtained by applying the method of moments [39] to Maxwell's equations. The application of the method of moments to Maxwell's equations results in the field theoretical foundation of the TLM method [40]. On the other hand, it has also been demonstrated in [40] that Yee’s FDTD scheme can be derived using the same approach with pulse functions for the expansion of the unknown fields.

The time-domain full-wave analysis is preferred when the transient analysis is required. However, it is not easy to incorporate the frequency-dependent interactions into the time-domain full-wave analysis. To treat the frequency-dependent effects such as dispersion, the convolution formula is involved in the time-domain and makes the time-domain full-wave analysis unrealistically slow.

1. References