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symmetry axis, this symmetry decomposition is trivial. Since we also want to use off-axis multipoles, we must decompose them, which leads to the concept of ring multipoles [6]. As an alternative to ring multipoles, complex origin multipoles [6] may also be used, but these are restricted to non-toroidal topology and have not been implemented in MaX-1. A spatial Fourier decomposition of the field is also used for periodic symmetries that are present in grating applications and in models of perfect photonic crystals. Since these models are infinite, they are idealizations that approximate realistic gratings and photonic crystals. There are several methods for handling periodic symmetries. First of all, one could proceed as for axisymmetric problems, i.e., one could implement symmetry adapted, periodic multipole expansions. This leads to convergence problems. Therefore, periodic symmetries are handled in MaX-1 with an alternative method, by the definition of periodic boundary conditions [6, 8]. These are very important for the modeling of gratings and photonic crystals. 3.1. Scattering and antenna A standard scattering problem is defined by a set of domains with the corresponding material properties and the incident wave. In most cases, the incident wave is considered to be a plane wave. It is well known that realistic waves generated by an antenna are not plane waves, but at a sufficient distance from the antenna and within a sufficiently small area, the antenna field may be approximated very precisely by a plane wave. As an example, we consider the scattering of a plane wave at an array of 6 times 6 cylindrical dielectric rods. This configuration may be considered as a finite 2D photonic crystal. Of course, the response of this structure depends not only on the orientation of the incident plane wave, but also on the material properties of the rods and on the size and spacing of the array. As one can see in figure 20, the block of 6 times 6 rods behaves very much as a square dielectric block, i.e., one can describe it quite well by some kind of "macroscopic" dielectric model that ignores the geometry of the crystal. However, when we sweep the frequency, we find that there are some frequency ranges where the block behaves like perfect conductor and totally reflects the incident wave for all angles of incidence as shown in Figure 21.

Efficient and Accurate Boundary Methods for Computational Optics

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Fig. 21: Finite photonic crystal consisting of 6 times 6 dielectric rods, illuminated by a plane wave. Frequency within the band gap. Left hand side: perpendicular incidence, right hand side: diagonal incidence.

Since we know that the incident plane wave is a part of an antenna field, we would like to know how to model an antenna, although this term usually is not used in optics. A standard antenna is a more or less complicated structure with some waveguide that feeds the antenna. Antenna design programs often do not explicitly model the antenna feed.

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Instead of this, some impressed current usually acts as the source of the field. This concept is not appropriate in optics. Here, we prefer the correct modeling of the feeding waveguide. As a simple example of an optical antenna, we can consider a finite photonic waveguide that contains a defect waveguide (see Figure 22). The defect waveguide acts as the antenna feed and it is considered to be infinitely long. MaX-1 allows us to compute the modes of such waveguides with different methods that will be outlined below. Once the modes of the waveguide are known, one can pack the expansions that describe the field of a mode in a connection. This allows us to truncate the infinite feeding waveguide and to model only a short section of it. Note that not only the mode that shall excite the antenna must be modeled, but also the modes that are reflected back into the waveguide feed. In addition to the guided modes, one may also have evanescent, higher order modes. When the waveguide section that is explicitly modeled is long enough, all evanescent modes will be damped sufficiently well in this section and need not be modeled by connections. On the other hand, in order to reduce the model size, one can also compute the relevant evanescent modes and pack them in connections.

Fig. 22: Time average of the Poynting vector field for a simple photonic crystal antenna fed by a single mode defect waveguide.

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3.2. Axisymmetric structures The handling of axisymmetric structures is relatively simple, when these structures have the same symmetry as one of the Fourier components of the angular symmetry decomposition. Such a symmetry is obtained when a plane wave is incident along the axis, but also when the structure is fed by a circular waveguide structure (for example, an optical fiber) along the axis. Typical examples for this are found in Scanning Nearfield Optical Microscopy (SNOM). Standard SNOMs use optical fibers with a metallic cladding and with a small tip or aperture. The fundamental fiber mode is the HE11 mode that has cos{cp) symmetry. This mode is much damped by the cladding [17]. In classical electromagnetics, a strong field confinement is obtained with simple multiconductor transmission lines. Such structures cannot be fabricated in the optical regime because metals lose their conductivity. However, at optical frequencies, some of the metals may be described by a complex permittivity with a negative real part, which can cause surface plasmons. When the imaginary part of the permittivity is small enough, one can use this for obtaining interesting waveguides and resonators. Figure 23 shows a concept study of a metallic resonator with a sharp tip illuminated by an axisymmetric light source. At some frequency range, an extremely bright spot is observed near the tip. The macroscopic model that was used shows no limitations

Fig. 23: Concept study of a resonant, metallic SNOM tip with an extremely localized electric field near the tip. The rotational symmetry axis is on the bottom of the figure. A special window with a finer grid and another scaling has been introduced around the tip in order to show the strength of the confinement together with the overall field distribution that is much weaker.

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concerning the spot size - provided that the radius of curvature near the tip is small enough. Therefore, the spot size will probably only be limited by atomic effects that cannot be handled with a pure Maxwell solver. The MMP solution of such axisymmetric models is very efficient, because only a 2D section of the 3D object is modeled. More precisely, only ID boundary lines are discretized. Therefore, one can work with small matrices of a few hundred unknowns even when the object is not small compared with the wavelength. For example, the model shown in Figure 23 has been computed with 252 unknowns. The average error is 0.05% and even at the most critical sections, the relative error is in the order of 1%. Since the MMP matrix is small, one can easily perform extensive studies that show the dependence on the frequency, on the illumination, on the material properties, and on the geometric shape. Another interesting structure is the axisymmetric, circular sinusoidal grating shown in Figure 24. As one can see, it generates a nice beam

Fig. 24: Axisymmetric, finite sinusoidal grating illuminated by a plane wave incident from the left hand side. Time average of the Poynting vector field. The symmetry axis is on the bottom of the figure.

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when it is illuminated by a plane wave along the axis. For this example, 521 unknowns were used and a relative error of 0.2% in the average along the boundary was obtained. The maximum relative error is in the order of 1%. Although axisymmetric problems can usually be solved with small MMP matrices it should be mentioned that the matrix setup time is much longer than the matrix setup time for cylindrical MMP models because the computation time of ring multipoles is much longer than the computation of cylindrical (2D) multipole expansions. 3.3. Periodic structures It has already been mentioned that periodic structures are handled with periodic boundary condition s in the MMP implementation of MaX-1. This allows one to separate a single cell of the structure. Once this has been done, the field inside the cell may be modeled exactly as the field in any other domain. For this purpose, one can use standard multipole expansions, Bessel expansions, and all other expansions available. A special modeling is required for gratings that may be periodic in one or two directions. Typically, one has a sandwich-like geometry. On top and bottom, one has a half space and the grating structure is somewhere in between as shown in Figure 25. When the grating

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is illuminated by a plane wave incident in the top half space, the field in this half space will also contain the reflected waves that may be expanded by special harmonic expansions, so-called Rayleigh expansions or Floquet modes. Similarly, the transmitted waves in the bottom half space may be modeled with Rayleigh expansions. Such expansions are available in the MMP code for 2D and 3D models. Figure 25 shows a 2D example. An example of a 3D model with a grating that is periodic in two perpendicular directions has already been given in Figure 18. Note that the half spaces of a grating must usually be separated from the grating structure itself by fictitious boundaries [6, 8] because the Rayleigh expansions alone are not complete when the half space is not limited by a planar boundary. Perfect photonic crystals are a class of periodic problems that is different from gratings because they cover the entire space. In fact, there is no excitation when perfect photonic crystals are studied, i.e., these structures are considered as eigenvalue problems whereas gratings are handled as scattering problems. Therefore, we first consider eigenvalue problems before we return to photonic crystals. 3.4. Guided waves and resonators Guided waves and resonators are typical eigenvalue problems. The resonator problem is simpler because the resonance frequencies are the eigenvalues that are characteristic for the modes, whereas the guided waves are characterized by the frequency dependency of the propagation constants of the modes. As we have seen, the PET is well suited for speeding up the calculation of the frequency dependency. Furthermore, we have seen that different methods may be used for defining eigenvalues. Within MaX-1, several methods are available, which provide a high degree of freedom. This is important because eigenvalue problems turn out to be very tricky when the geometry is not very simple, when many modes are present, and when losses are present. In order to illustrate, we consider a so-called holey optical fiber consisting of 6 circular holes on a hexagonal lattice. Figure 26 shows the first quadrant of this structure. The fundamental mode of the holey fiber resembles the HE11 mode of the circular step index fiber, but the classification of the modes is extremely difficult because the holey fiber does not exhibit rotational

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symmetry. A partial classification might be obtained from the symmetry that is still there: The holey fiber has 6 symmetrical axes, i.e., diedric D6 symmetry. The symmetry decomposition for D6 is not very difficult, but it is not trivial as well, because the corresponding symmetry group is two-dimensional. Furthermore - as most codes for computational electromagnetics - the MMP implementation of MaX-1 does not allow one to take D6 symmetry into account. It only considers up to 2 perpendicular symmetrical axes, i.e., D2 symmetry. When we only consider D2, we obtain an incomplete symmetry decomposition, but this will still allow us to distinguish four classes of modes with different symmetrical properties. This considerably simplifies the detection for the following reasons. Assume that 100 modes are present. Without symmetry consideration, one will have to perform a rough search with at least 10000 search points, because some of the modes can have very similar eigenvalues. When D2 symmetry is considered, each of the four classes of modes will have about 25 modes and the distances between them will be considerably bigger. Therefore, 4 times 1000 search points should be sufficient. Furthermore, the size of the matrix that describes the modes is considerably smaller (N/4 times M/4 instead of N times M) when D2 symmetry is considered. Therefore, the computation time for the rough search will be reduced by a factor of approximately 160.

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As one can see from Figure 26, the eigenvalue search function exhibits extremely sharp minims that characterize the modes mainly near the right hand side of the diagram. In order to accurately compute the field of the corresponding mode, we must explore the corresponding minims with a high precision. For example, the rightmost minimum that corresponds to the fundamental mode goes down to 1E6. The minims that characterize higher order modes are more flat and high. Although this simplifies the search for the corresponding modes, it also indicates that an insufficient accuracy will be reached because the orders of the multipoles and Bessel expansions are not high enough for a good approximation of these modes with their complicated field patterns. 3.5. Perfect photonic crystals Perfect photonic crystals are structures that are periodic in up to three directions in space. For reasons of simplicity, we focus on cylindrical 2D crystals that are periodic in two directions of the xy plane that is perpendicular to the cylinder (z) axis. Such crystals usually consist of dielectric rods or holes in a dielectric, but metallic rods may also be considered [15, 16]. For the analysis of photonic crystals, methods known from crystallography are applied. Its special symmetries are first explored. For the MMP analysis, fictitious boundaries are introduced for separating a single cell of the original lattice. Along these boundaries, periodic boundary conditions are imposed. Here, it is important to note that the fictitious boundaries may be straight lines - which seems to be natural - but curved lines are admitted as well. If one already has some idea about the field pattern of the mode to be analyzed, it is reasonable to define the fictitious boundaries in an area where the field is relatively smooth because this will minimize the model size. As shown in Figure 27, only a single original cell of the original lattice must be modeled explicitly. Since this cell often exhibits additional symmetrical axes, it would be possible to model only a part of it and to apply appropriate symmetry decomposition, but this has not been implemented in the current MMP version. As a consequence, one may encounter degenerate modes that cause some numerical problems.

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In the second step, the reciprocal lattice space is introduced which is essentially obtained from a spatial Fourier analysis. This space is illustrated in Figure 28. It is spanned by the components of the wave vector. Because of the symmetry, it is sufficient to analyze the modes only over a finite part of the reciprocal space, the 1st Irreducible Brillouin Zone (IBZ) as shown in Figure 28. For the most frequently used square and triangular (hexagonal) lattices, the IBZ is a triangle.

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In order to compute the band gap that is of highest interest for photonic crystals, it is sufficient to explore the borders of the IBZ only, i.e., to find the resonance frequencies along the lines in Figure 28 from r to X, from X to M, and from M back to r. Figure 29 shows a typical band diagram. As one can see, the band gaps are zones or frequency ranges (vertical direction in Figure 27) where no mode exists. When a finite photonic crystal is illuminated by a wave incident from an arbitrary direction, it will totally reflect the wave. This interesting and useful effect is only obtained when the dielectric contrast between the rods and the background si sufficiently high and when the rods are sufficiently big. Furthermore, the geometric shape of the rods and the symmetry of the lattice play important roles. Since the band gaps define the working area for photonic crystals, the main reason for the computation of the band diagrams is to find crystals with a sufficiently wide bandgap around the desired frequency range. This is a rather difficult optimization task, especially because the computation of the band diagram exhibits several numerical difficulties. Fortunately, one usually works in the lower band gap. In order to find it, only the lowest order modes need to be considered. These modes have a relatively simple field and can be computed with relatively low computational effort. MaX-1 provides all tools required for the

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efficient computation of band diagrams and for dealing with numerical difficulties. 3.6. Waveguides in photonic crystals Although it is nice to fabricate a photonic crystal that totally reflects all waves within a certain frequency range, defects make photonic crystals much more attractive and promising. In fact, photonic crystals were first proposed as an optical counterpart of semiconductor crystals [19]. As we know, doping makes semiconductors really interesting. Since the cells of photonic crystals are much bigger than the atoms of a semiconductor, one can "dope" photonic crystals very precisely by changing the geometry and material properties of one or several cells. For example, when a photonic crystal consists of a set of dielectric rods on a square lattice, one can remove all rods along a line [20, 21]. As a result, a simple defect waveguide is obtained as shown in Figure 22. Note that it is very easy to obtain many different types of waveguides simply by modifications of the cells of a photonic crystal along one or several parallel lines [22]. This gives one much freedom in the design of waveguides, but it also poses a problem to design and optimize waveguides with desired properties, which is a hard synthesis problem. The standard approach to compute waveguides in photonic crystals is the so-called super cell approach [23, 24] that considers an array of parallel waveguides instead of a single one. When the waveguides are separated by sufficiently many cells of the photonic crystal and when one operates sufficiently well within the band gap, there is almost no interaction of the waves in neighboring waveguides. Therefore, one essentially obtains the same field distribution as in the single waveguide. The set of infinitely many parallel waveguides may be considered as a bigger periodic structure with an original cell that contains several cells of the former, undisturbed photonic crystal. Figure 30 shows an example of a super cell with a defect waveguide in the center and four rods on each sides of the channel, i.e., neighboring channels are separated by 8 rows of rods. As one can see in Figure 30, each of the modes of the undisturbed crystal on both sides of the band gap splits in several modes. The number of modes increases with the size of the super cell, i.e., with the number of rods on each side of the channel. Therefore, the computational effort is drastically increased, which makes the super cell method inefficient.

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Beside the split modes, one also observes a completely new mode within the original band gap in Figure 30. This is the defect waveguide mode. Note that this mode is not completely within the band gap, but it covers a wide area of it. Depending on the model parameters, one can obtain single mode waveguides as in Figure 30, but multimode waveguides as well. Although the super cell method allows one to use the same procedures as for the analysis of perfect photonic crystals, this method is obviously neither very realistic nor numerically efficient. A more realistic waveguide model would consist of a channel with a finite number of rows of rods on both sides, i.e., one might simply remove the fictitious boundaries on top and bottom of the super cell shown in Figure 30 and keep the left and right fictitious boundaries. When one does this, the structure is no longer periodic in the vertical direction, but it is still periodic in the horizontal direction. Thus, one obtains a one-dimensional lattice space and a one-dimensional reciprocal space as well. The reciprocal space now essentially is the space of the propagation constant of the waveguide, i.e., the wave number in horizontal direction. This makes the eigenvalue search much easier. When one removes the fictitious boundaries that separate the super cell from the neighboring cells with the parallel neighboring waveguides, one should recognize that the resulting waveguide with finite walls may

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radiate in the upper and lower half space. When the number of rows of rods on both sides of the channel is big enough, one may expect that the radiation is weak and may be neglected. In order to obtain information about the radiated power, we must take it into account. Within our MMP model, we can easily do this by a model similar to the grating model that separates the upper and lower half space by fictitious boundaries. Therefore, we can re-introduce the fictitious boundaries on top and bottom of Figure 30. Instead of imposing periodic boundary conditions, we match all field components and model the radiated field in the top and bottom half space with simple multipole expansions. The radiation losses have an important consequence: the propagation constant now must be complex even if all dielectrics are loss-free. This requires a complex eigenvalue search instead of a real search. When the radiation loss is not very high, one can first ignore it and perform a real eigenvalue search. As soon as the field of the mode is known, one can also compute the radiated power and the total power transmitted along the channel. From these data, one can easily approximate the attenuation constant of the mode. In order to illustrate the procedure, we consider a defect waveguide with only one row of rods on each side of the channel as shown in Figure 31. Obviously such a waveguide must exhibit the strongest possible radiation. Therefore, we perform a complex eigenvalue search. In order to get an overview over the eigenvalue search function in the complex

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plane (see Figure 32), we start the rough search on a rectangular grid defined on the complex plane. When we want to have a fine resolution, this is time-consuming. Note that this is not necessary in general because there are not many different eigenvalues of interest, i.e., eigenvalues with a reasonably small propagation constant. Therefore, we could also start with a rough search along the real axis and perform the fine search from the resulting start points into the complex plane, which would be numerically much more efficient. As soon as we have found the desired eigenvalue that characterizes the fundamental (radiating) mode, we may compute and plot its field (see Figure 31) and we may track its frequency dependency in the complex plane using the PET (see Figure 32). As one can see, the attenuation constant (vertical direction in Figure 32) drastically grows when the frequency is reduced and it is quite big when the lower end of the band gap is reached. Furthermore, we can trace the mode also outside the band gap although this makes not much sense for practical applications. However, we observe that the behavior of these photonic crystal waveguide modes is similar to the behavior of lossy fiber modes [17] although the former exhibit radiation loss whereas the latter exhibit Ohmic loss in the coating. From the practical point of view, it is important to know that one should not operate near the lower limit of the frequency band in order to avoid strong attenuation due to strong

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radiation loss. Of course, we may drastically reduce the radiation loss by adding one or several rows of photonic crystal cells on both sides of the channel (see Figure 32). When the radiation loss is not very high, the complex eigenvalue search can be avoided. When we first assume that the attenuation constant is zero, we can perform a real eigenvalue search (along the real axis of the complex plane) and obtain a real valued approximation of the complex propagation constant, i.e., an approximation of the phase constant. As one can see from Figure 33, a good approximation is obtained even in the worst case with a single layer of rods on both sides of the channel. As soon as the phase constant is known, one can also compute the electromagnetic field and from this, one can estimate the power loss along the guide due to radiation. From this, one can easily obtain an estimate of the attenuation constant that is also shown in Figure 33. As one can see, the estimation is quite good - even for the worst case - over the entire frequency range of the band gap. Obviously, the error is higher for lower frequencies. It should be mentioned that the defect mode considered here has a cutoff near the lower end of the band gap. One can see this from models with a sufficiently large number of layers on both sides of the channel. Note that the cutoff frequency of any waveguide mode is strictly defined only in the loss-free case. When Ohmic losses or

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radiation losses are present, some sort of smooth transition is observed as shown in Figure 32. 3.7. Discontinuities in photonic crystals Although it is important to have waveguides in photonic crystals, the existence of such waveguides would not make the photonic crystal concept sufficiently attractive for fabrication although such waveguides may be smaller than traditional waveguides used in integrated optics. The photonic crystal concept shows its huge potential when one studies discontinuities of waveguides. Theoretically, periodic discontinuities might be studied with the super cell method. As we have already seen, this method is not efficient at all. Therefore, we search for an alternative approach. Although finite photonic crystal waveguide discontinuities are somehow different from the discontinuities of traditional waveguides, we may use the same concept for a study with MMP [25, 26]: 1) Essentially all waveguide discontinuity problems consist of at least one input waveguide and usually of one or several output waveguides. These waveguides are assumed to extend to infinity. The modes in these waveguides can be analyzed using the MMP eigenvalue solver. 2) The waveguide sections may be separated with fictitious boundaries from the discontinuity region. The fictitious boundaries play the role of reference planes known from measurement. Let us call them ports. In each port one has a finite number of propagating modes and an infinite number of evanescent modes. The field in the discontinuity region can be modeled by a standard MMP expansions. It is linked to the waveguide modes by the fictitious boundaries at the ports. The field inside the discontinuity region is excited by the incoming modes and it excites the outgoing modes. In MMP, fictitious boundaries and connections allow one to easily link the field with the modes (see Figure 34). Since the evanescent modes are excited in the discontinuity region and decay exponentially, one can omit them when the waveguide port is sufficiently far away from the discontinuity. Moving ports away from the discontinuity increases the size of the discontinuity region that is explicitly modeled and this considerably increases the computation time. Therefore, it is sometimes more reasonable to move the ports close to the discontinuity region, where the lowest evanescent modes are still strong. If this is done, the field in the waveguides must be approximated by a

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superposition of the Ng guided waves and of the first Ne evanescent modes. As we have seen, the MMP eigenvalue solver can find all guided waves of an infinite waveguide, even when it is lossy. Since MMP can search eigenvalues in the complex plane, it also can find evanescent modes. Once a waveguide mode is known, one can pack the expansions that model its field in a connection. One then can model the field in each port with Ng + Ne connections. This allows one to match the field at the ports with a high precision in such a way that almost no reflections at the ports are observed. Figure 35 illustrates that for a simple case. As soon as one knows how to handle waveguide discontinuities, one can start designing interesting structures in photonic crystals, for example, multiplexers [27], resonators, filters [28], etc. It is well known that the device synthesis is much more demanding than the analysis of an existing device or structure. The main problem in the photonic crystal design is that almost no design rules are currently available. For example, we know that we can obtain filters, but we don't have rules how to design a filter with specified characteristics. In order to illustrate a possible design procedure, we now consider a filtering T junction with one input port and two output ports [27]. We would like to design this

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Fig. 35: Poynting vector field of a defect waveguide with two sharp 90° bends.

structure in such a way that it transmits almost all energy from the input port to the first output port at one frequency / , and to the second output port at another frequency f2. First, we select a perfect photonic crystal with a sufficiently wide bandgap that contains both frequencies// and/}. Since we would like to have waveguides in two perpendicular directions, we use a square lattice. Furthermore, we consider a crystal consisting of dielectric rods rather than of holes in a dielectric. Note that the latter would be better for some practical reasons, but the former is numerically simpler and we already used it in the examples shown above. In the second step, we design three different single mode waveguides that shall be used for the three ports. When the input waveguide propagates at all frequencies of the entire band gap, whereas the first output port mode only propagates within a limited frequency band around // but not at f2, we can expect that most of the power is transmitted to the first output port at //. Similarly, we can design the second output waveguide in such a way that most of the power is transmitted to the second output port at /}. Figure 36 shows results obtained with this concept. Although the design with three different waveguides is promising, we do not want to have structures with different waveguide types. When we want a filtering T junction with three identical waveguide ports, we can use the model above and connect the two output waveguides with

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Fig. 36: Filtering T junction with three different waveguides. Top: Waveguide mode I) red: input port (from top), 2) blue: left port, 3) green: right port. Bottom: Poynting vector plots for the two transmission frequencies.

waveguides of the same geometry as the input waveguide after a certain length. It is obvious that this will cause some additional reflections at the transitions between the different waveguides and that the quality of the T junction will considerably depend on the lengths of the two output waveguides. Since the main goal of photonic crystals is miniaturization, we try to make the T junction as small as possible. Figure 36 shows such a T junction that is obtained after some optimization of the two output sections. The main ideas of this optimization are that the propagation constants of the modes of the input and first output section should be the

CK Hafner & J. Sraajic

124

I bandgap [Right propagation frequency range [yellow area ) O O O O O O

C

J O O O O O O O O O O O O o o o o o o o o o * r, n O o O n O

o o o o o a o o o o o o o l o o

o

o o | o o

0

o o o o o

Fig. 37: Filtering T junction with identical waveguide ports. Top: Waveguide mode 1) red: input port (from top), 2) blue: left section, 3) green: right section. Bottom: Poynting vector plots for the two transmission frequencies.

same at//. Furthermore, the group velocities of these modes should be similar at//. The same shall hold for the second output section a t / . Finally, we know that the evanescent field of the undesired mode will reach far into the wrong output port when one operates close to the cutoff frequency. Therefore, the cutoff frequency of the second output mode should not be close t o / and vice versa. Figure 37 shows the resulting structure. With this simple design, a rather good performance (approximately 89% transmission to port 1 f o r / and 93% to port 2 f o r / ) is obtained with a simple tuning of the waveguide sections. In order to obtain even better performance, one must either increase the size of the T junction or one must perform more comprehensive optimization of the entire T junction area, which is numerically very time consuming.

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0 O O B

o o o o o o 0 0 0

Fig. 38: Compact filtering T junction. Top: Waveguide mode 1) red: input port (from top), 2) blue: left section, 3) green: right section. Bottom: Poynting vector plots for the two transmission frequencies.

Obviously, the left channel that is similar to a coupled cavity waveguide structure is geometrically more complicated and also considerably longer than the right channel that simply consists of three thick rods. In order to obtain a more compact design, we replace the left waveguide section by a waveguide section similar to the right waveguide section as shown in Figure 38. The new structure is considerably more compact, but its performance is considerably worse. We now try to improve it with a simple optimization of the transition region. In fact, there would be many model parameters (size and positions of all rods in the transition area) that might be optimized, but such an optimization in a high-dimensional parameter space would be too time-consuming for our personal computers. Therefore, we select only two important parameters and optimize them. These parameters are the distances of the rods in the two

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Fig. 39: Compact filtering T junction optimized. Poynting vector plots for the two transmission frequencies.

output sections. Since the minimum reflection coefficients and the maximum transmission coefficients for the two output ports at// and/ 2 are obtained for different values of the two distances, we must find some compromise. The resulting design is shown in Figure 39. This optimization allows us to increase the transmission from 63% (Fig. 38) up to 88% (Fig. 39) for both channels, i.e., the new T junction is almost as good as the one shown in Figure 37, but it is much more compact and easier to fabricate. When we optimize a photonic crystal structure, we see that some of the model parameters may be rather critical. In our filtering T junction, the size and locations of the rods in the transition area are quite critical. This means that a precise fabrication is required. Otherwise the performance of the junction might be considerably worse. Although this may cause problems for the fabrication, it is also interesting for tuning. It is well known, that photonic crystals may be tuned [29, 30], for example, with an external electrical field, when appropriate materials are introduced. When the material properties in the transition area are modified or when the sizes of the rods in this area are modified, essentially the same effects are obtained. Therefore, it should be possible to tune the properties of the filtering T junction with external electrical field, i.e., we may obtain also optical switches [31] with a similar, compact T-junction design. Conclusions We have presented a powerful method for the accurate and efficient fullwave analysis of electromagnetic waves that is very useful for the analysis and design of novel structures for ultra dense integrated optics, namely for photonic crystal structures. Furthermore, we have presented

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several technique for speeding up the analysis of sequences of similar models and for obtaining efficient and precise eigenvalue solvers. Due to its high reliability and accuracy, our code is very well suited for future optimizations of novel photonic crystal structures. Acknowledgments This work was supported by the Swiss National Science Foundation. References 1. Ch. Hafher, (in German) (Diss. ETH, No. 6683, Zurich, 1980). 2. G. Mie, Annalen der Physik. , 2 (1900). 3. G. Mie, Beitrdge zur Optik triiber Medien, speziell kolloidaler Metallosungen. (Ann. Phys. 25, 377-445, 1908, An English translation is available as G. Mie, Contributions to the optics of turbid media, particularly of colloidal metal solutions, Royal Aircraft Establishment library translation 1873. Her Majesty's Stationery Office, London, 1976). 4. I. N. Vekua, New methods of solution of elliptic equations. (Russian) (Gostekhizdat, Moscow-Leningrad, 1948). 5. I. N. Vekua, New methods of solution of elliptic equations. (North-Holland Publ. Co., Amsterdam, 1967). 6. Ch. Hafner, Post-modern Electromagnetics Using Intelligent MaXwell Solvers. (John Wiley & Sons, 1999). 7. Ch. Hafner, The GGP Code Web Page. (IFH, ETH Zurich), http://alphard.ethz.ch/ggp . 8. Ch. Hafner, MaX-l: A Visual Electromagnetics Platform. (John Wiley & Sons, 1998). 9. F. G. Bogdanov, D. D. Karkashadze, and R. S. Zaridze, Generalized Multipole Techniques for Electromagnetic and Light Scattering, (edited by T. Wriedt, pp. 143-172, Elsevier, Amsterdam, 1999). 10. G. Tayeb, The Method of Fictitious Sources Applied to Diffraction Gratings. (Applied Computational Electromagnetics Society Journal, Vol. 9, No. 3, 1994). 11. V. Rokhlin, Rapid solution of integral equations of scattering theory in two dimensions. (J. Comp. Phys. 86, 414-439. 1990). 12. G. H. Golub and C. F. Van Loan, Matrix Computations. (Johns Hopkins University Press, Baltimore, 1996). 13. M. R. Hestenes, E. Stiefel, Methods of Conjugate Gradients for Solving Linear Systems. (Journal of Research of the National Bureau of Standards, Vol. 49, No. 6, December 1952).

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14. J. Smajic, Ch. Hafher, D. Erni, Automatic Calculation of Band Diagrams of Photonic Crystals Using the Multiple Multipole Method. (ACES Journal, to be published). 15. K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, K. Hirao, Photonic bands of Metallic Systems. I. Principle of Calculation and Accuracy. (Phys. Rev. B, 64, pp. 045116,2001). 16. E. Moreno, D. Erni, Ch Hafher, Band Structure Computations of Metallic Photonic Crystals with the Multiple Multipole Method. (Phys. Rev. B, 65, pp. 155120, 2002). 17. L. Novotny, Light Propagation and Light Confinement in Near-Field Optics (Diss. ETH,No. 11420, Zurich, 1996). 18. Ch. Hafner, J. Smajic, The Computational Optics Group Web Page. (IFH, ETH Zurich), http://alphard.ethz.ch . 19. E. Yablanovich, Inhibited SpontaneousEemission in Solid-StatePhysiscs and Electronics. (Phys. Rev. Lett. 58, pp. 2059-2062, 1987). 20. E. Centeno, D. Felbacq, Guiding Waves with Photonic Crystals. (Opt. Commun. 160,57,1999). 21. H. Benisty, Modal Analysis of Optical Guides With Two-Dimensional Photonic Band-Gap Boundaries. (J. Appl. Phys. 79, pp. 7483-7492, 1996). 22. M. Loncar, J. Vuckovic, A. Scherer, Methods for controlling positions of guided modes of photonic crystal waveguides. (J. Opt. Soc. Am. B 18, pp. 1362-1368, 2001). 23. K. Sakoda, Optical Properties of Photonic Crystals. (Springer, Berlin, 2001). 24. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Molding the Flow of Light. (Princeton University Press, 1995). 25. E. Moreno, D. Erni, Ch. Hafher, Modeling of Discontinuities in Photonic Crystal Waveguides With the Multiple Multipole Method. (Phys. Rev. E, 66, pp. 036618, 2002). 26. E. Moreno, MMP Modeling and Optimization of Photonic Crystals, Optical Devices, andNanostructures. (Diss. ETH, No. 14553, Zurich, 2002). 27. J. Smajic, Ch. Hafher, D. Erni, On the Design of Photonic Crystal Multiplexers. (Optics Express, to be published). 28. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, Channel Drop Filters in Photonic Crystals. (Opt. Express 4, pp. 4-10, 1998). 29. K. Busch, S. John, Liquid-Crystal Photonic-Band-Gap Materials: the Tunable Electromagnetic Vacuum. (Phys. Rev. Letters, 83, pp. 967-970, 1990). 30. A. Figotin, Y. A. Godin, Two-Dimensional Tunable Photonic Crystals. (Phys. Rev. B, 57, pp. 2841-2848, 1998). 31. R. L. Sutherland, V. P. Tondiglia, L. V. Natarajan, S. Chandra, Switchable Orthorombic F Photonic Crystals Formed by Holographic polymerisation-Induced Phase Separation of Liquid Crystal. (Optics Express, 10, pp. 1074-1082, 2002).

FINITE ELEMENT MODELING OF PERIODIC STRUCTURES

Zheng Lou and Jian-Ming Jin Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2991 E-mail: [email protected] Periodic structures have a variety of important applications in modern technologies and engineering due to their unique electromagnetic properties. Commonly used periodic structures include frequency selective surfaces, optical gratings, phased array antennas, photonic bandgap materials, and various metamaterials. The analysis of periodic structures has always been an important topic in computational electromagnetics. In this chapter, we describe an accurate and efficient numerical analysis, based on a higher-order finite element method (FEM), for characterizing the electromagnetic properties of periodic structures. Based on the Floquet theory, periodic boundary conditions and radiation conditions are first derived for the unit cell of a periodic structure. The FEM is then applied to solve Maxwell's equations in the unit cell. To enhance the accuracy and efficiency of the FEM, curvilinear elements are employed to discretize the unit cell and higher-order vector basis functions are used to expand the electric field. The asymptotic waveform evaluation (AWE) is implemented to perform fast frequency and angular sweeps. To demonstrate the capability of the proposed FEM, we apply it to the analysis of periodic absorbers, frequency selective structures, and phased array antennas. For the antenna analysis, a rigorous waveguide port condition is developed to accurately model the antenna feed structures. In all the cases studied, satisfactory results are obtained.

1. I n t r o d u c t i o n Periodic structures have been extensively used in electromagnetic engineering. T h e periodicity in the geometry is often exploited to achieve certain desired electromagnetic properties, such as frequency selective behaviors,

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which are not obtainable in the case of a single element. Many microwave and optical devices, such as anechoic chamber absorbers, frequency selective structures (FSS), and phased array antennas, fall into this category. Numerical analysis of periodic structures has been carried out using a variety of numerical methods, such as the moment method (MoM), the finite-element method (FEM), and the finite-difference time-domain (FDTD) method. Among these methods, the FEM excels at modeling complicated geometry and material inhomogeneity. The method is also versatile in its ability to incorporate different types of boundary conditions and different excitation modes without significantly affecting its formulation. The FEM modeling of periodic structures has been reported in literature for both scattering [l]-[5] and radiation [6]-[8] analyses. In this chapter, we describe a robust, higher-order FEM formulation to model infinitely periodic array structures. By imposing appropriate radiation boundary conditions and periodic boundary conditions, the computation domain is confined to a single unit cell in the infinite array. The unit cell interior region is discretized with curvilinear tetrahedral elements for better geometry modeling. The electromagnetic field is then expanded with higher-order vector basis functions, which are more efficient than the conventional zeroth-order basis functions. For broadband calculations, the asymptotic waveform evaluation (AWE) technique can be combined with the FEM to perform fast frequency and angular sweeps. For radiation analysis, an accurate coaxial feed modeling has been implemented, employing a rigorous waveguide port condition. For the purpose of validation, numerical examples are given for both scattering and radiation analyses for various element configurations. The agreement with previously published data and measured results demonstrates the accuracy and capability of the proposed FEM formulation. 2. Formulation This section describes various aspects of the FEM formulation for the analysis of periodic structures. The fast sweep techniques are also outlined at the end of this section. 2.1.

Discretization

The structures under consideration are planar arrays, infinitely periodic in the x and y directions. Since all the elements in the array are identical, the FEM analysis is applied only to a single element, or a unit cell, as shown

Finite Element Modeling of Periodic

Structures

131

in Fig. 1. We start from Maxwell's equations for the electromagnetic fields inside the computation domain:

/ //

X

/

r

7 //

/

/

v

/

7

..... . / / / / h / ..../_ ~T 7 / / '// / / / / / y/ / / 7 / /" / / / / /

/

./ ./

/ /

/

/ /

/

/

J...

Fig. 1.

/ / Unit /

cell

Unit cell in an infinite periodic structure.

V XE =

-jcdfiofirll

(1) (2)

V x H = jwe 0 e r E + J.

Eliminating H in the two equations above results in the wave equation for E: V x I —V x E

k20erE =

-jk0ZoJ.

(3)

Assuming that E satisfies certain boundary conditions on the surface S that encloses the computation domain V, it can be shown that the original problem is equivalent to the following variational problem [9]: 6F(E) = 0

(4)

where

^-WiZ

-(V x E ) - ( V x E ) - f c ^ e r E - E dV

° °JJL

+jk<

E-Jd^

+ surface integral terms

(5)

where the surface integral terms are the results of imposing corresponding boundary conditions. In this subsection, we only consider the interior part

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of the problem. The boundary condition terms will be discussed in the following subsection. To solve the variational problem (4) numerically, the entire computation domain V is discretized into small elements. In the FEM discretization, commonly used elements are triangular elements for 2-D problems and tetrahedral elements for 3-D problems. To faithfully model curved surfaces, curvilinear elements are often utilized. For the curvilinear elements, the evaluation of the integrals in (5) is facilitated by mapping the integrals onto a corresponding straight element with unit length in a new coordinate system. For triangular elements, the new coordinate system is denoted as the £r) coordinate and the mapping is given by 6

* = $">?(£, j ^

(6)

2/ = E ^ > ^

(7) ' dx dy

dxdy = | J\d£drj

[J] =

(8) dx dy . drj dr) where Xi and yi are the coordinates of the ith node in the xy coordinate system and Nf{£,rj) are scalar shape functions defined in the £rj coordinate system. The mapping for tetrahedral elements can be defined in a similar fashion with 10

* = YlNifoTl>Oxi

(9)

i=l 10

(10) i=l 10

(ii) ' dx dy dz

~bl;~dl,dl, dxdydz = \J\d^drjd^

[J] =

dx dy dz dr] dr/ dr} dx dy dz

^Cd~cd~c

(12)

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Structures

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Using the mapping defined in (6)-(12), the integration is carried out in the new coordinate system, where it can be evaluated easily using GaussianLegendre quadratures. It is also more convenient to define all the basis functions in the new coordinate system. The next step involves expanding the electromagnetic fields in terms of basis functions in each element. In the context of 3-D problems, it is more convenient to use vector basis functions instead of scalar basis functions. For tetrahedral elements, the vector basis function associated with the ith edge is given by Wf^L^VL^-^VLfJi?

(13)

where i\ and ii denote the two vertices associated with the ith edge and If is the length of the ith edge. Further, L% (k = 1,2,3,4) are the nodebased basis functions associated with the four vertices of the tetrahedral. It can be demonstrated easily that Wf automatically satisfies the divergence condition. Thus, the solutions obtained by using vector basis functions are free of spurious solutions. Also, it is easier to model dielectric interfaces and conducting edges and tips using vector basis functions than using scalar basis functions. The vector basis functions defined in (13) are zeroth-order interpolating functions. Higher-order basis functions can be constructed systematically by multiplying the zeroth-order interpolating functions with Lagrange interpolator polynomials [10] given by

Nj$=7if+2(£i)^r2(6)^r2(6)£r2(£4)will2

(u)

where (11,12) are the combinations of two integers from (1,2,3,4) and is and «4 are two integers among (1,2,3,4) other than i\ and ig. Also, i, j , k, and I satisfy i+j + k + l=p+2 where p is the order of basis functions, 7 is a normalization constant, W ; ^ are zeroth-order basis functions, and Pf (£) are the shifted Silvester polynomials defined by P +2

? (0=

(- 1 !)! 3f[[(P + 2)e-i].

(15)

The detailed discussion of higher-order basis functions can be found in [10]. It has widely been accepted that higher-order basis functions are more accurate and efficient in modeling the actual field solution for electromagnetic problems. As an example, Fig. 2 shows the relative error versus the number of total unknowns for using different order basis functions in a 2-D eigenvalue problem. The eigenvalue problem is to determine the cutoff

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wavenumber of the TE22 mode propagating in a rectangular waveguide. The relative error is obtained by comparing the numerical and analytical cutoff wavenumbers. It is easily seen from Fig. 2 that for a given number of unknowns, the higher-order basis functions produce a higher accuracy than do the lower-order basis functions. Moreover, different error curves in Fig. 2 exhibit different slopes, indicating that the relative error decreases much faster by choosing higher-order basis functions. In a general sense, one can always achieve a higher accuracy by using higher-order basis functions. Practically, however, the choice of interpolating order depends on specific problems. It is advisable to choose the order of basis functions such that it is not much higher than the order of geometrical transformation.

10 -

i" 10

*

\ - * - Oth order - O 1st order • A • 2nd order

.

\ A 10'

10°

Number of Unknowns Fig. 2. Relative error in computing TE22 cutoff wavenumer of a 3 x 2 cm rectangular waveguide.

Once the basis functions are determined, the electric (or magnetic) field within each element can be expanded as Ee = ^ N ? B ?

(16)

i=i

where n is the number of interpolating points (depending on the order of basis functions) within each element. Substituting (16) into (5), one arrives

Finite Element Modeling of Periodic

Structures

135

at the elemental matrices [Ke] with the entries given by — (V x Nf) • (V x N?) - AgerN? • N dV and the right-hand-side (RHS) known vector {be} given by b\ = -jfcoZo ^ y

N? • JdV.

(17)

The [Ke] and {be} are then assembled to the system matrix [K] and system known vector {b}, respectively, according to the connectivity of the FEM mesh. The result of subdomain discretization, basis function expansion, and finite element assembling is a set of linear equations in the matrix form of [K] {E} = {b}.

(18)

Both direct and iterative methods can be employed to solve the linear system equations. Popular iterative methods include the conjugate gradient (CG) method and the generalized minimal residual (GMRES) methods. For direct methods, the frontal and multifrontal algorithms [11] are suitable solvers for linear systems resulting from the finite element assembling procedure. In the finite element assembling procedure, a large number of elemental equations are decoupled. Taking advantage of this fact, the frontal method performs Gaussian elimination as soon as an equation is completely decoupled from the rest of the equations. The multifrontal method is an extension of the frontal method, which employs multiple frontal matrices. The advantage of the frontal/multifrontal method is its capability to deal with arbitrary sparsity pattern, making it far more efficient than the standard LU decomposition method. A popular multifrontal package is called UMFPACK a , which is a sparse linear solver based on the unsymmetric-pattern multifrontal method. 2.2. Boundary

Conditions

Consider the unit cell in an infinite periodic structure, as shown in Fig. 3. The interior volume, denoted here as V, is enclosed by four side surfaces, a top surface, and a bottom surface. It may contain arbitrary dielectric and conducting structures. The four side surfaces Sxi,SX2, Syi, and Sy2 are located at x = 0, x = Tx,y = 0, and y = Ty, respectively, where Tx and a

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Ty are periodic lengths in the x and y directions. The top surface St is the interface between free-space and the unit cell region. The bottom surface Sb is usually a ground plane. It may also contain waveguide apertures Sw that provide excitation for the radiation case. In such a general configuration, four kinds of boundary conditions are involved. On the four side surfaces, periodic boundary conditions are imposed, relating the fields on the opposite side surfaces. On the top surface, a periodic radiation boundary condition is imposed, which simulates the radiation towards the free space in the presence of an infinite array. If a waveguide is present in the structure, then a waveguide port condition is imposed on Sw. Finally, on conducting surfaces, a perfectly electrically conducting (PEC) boundary condition is enforced explicitly as a homogeneous Dirichlet boundary condition.

Fig. 3.

Illustration of boundaries on a unit cell.

Periodic boundary conditions can be derived directly from the Floquet theorem. In accordance with the Floquet theorem, the electromagnetic fields inside and above the periodic media satisfy E(x + mTx,y + nTy) = E{x,

mk T +nk

y)e^

"-

v^y)

(19)

H(x + mTx, y + nTy) = H(x, j,)(J'(TOfc-*r*+nfc»'r»)

(20)

Finite Element Modeling of Periodic Structures

137

kxi = k0 sin 6l cos <j)1

(21)

where

l

1

kyi = k0 sin 9 sin

(22)

where (0l, <j)1) are the incident or scan angles. To facilitate the implementation of periodic boundary conditions, identical surface meshes are created on the opposite side surfaces. Then for each unknown Ei on one side surface, a corresponding unknown Ej is identified on the opposite surface which has the same relative position as Ei. Applying the Floquet theorem (19), one can easily obtain the following relationship between Ei and Ef Ej = Ei e J '*«

(23)

where ^ y is a phase shift term given by

*«= <

kxilx

Vi/j € oxi,

Ej € oX2

kyiTy

VEiGSyU

Ej E S y2

kXiTx + kyiTy

vEi G Sxi n Syi,

(24)

Ej G Sx2 n Sy2-

In the matrix context, (23) is enforced explicitly as an inhomogeneous Dirichlet boundary condition. T h a t is, for each unknown pair {Ei,Ej): Ej is eliminated; the matrix entries associated with Ei are modified as Ku = Ku + Kji ( J * "

(25)

Kik = Kik + Kj,e?(*i>-*")

(26)

for all Ei £ 5X2 U 5^2, and

for all Ei € 5X2 U Sy2 which is related t o Ek by phase shift ^ki', and finally, t h e RHS vector entry associated with Ei is modified as bi = bi + bje^.

(27)

T h e functional given in (5) contains a surface integral term over the top surface St with a form given by 5 ( E ) = -jk0Z0

jf

M • HdS

(28)

where M = — n x E is the equivalent magnetic current on St- Hence, there is a need t o derive an equation t h a t relates the magnetic current M and the magnetic field H . In order to do t h a t , we first assume t h a t a ground

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plane is placed on St- From the equivalence principle, the total magnetic field above St can be written as H(r) = H i n c (r) + H r e f (r) + H s c a t (r) lnc

(29)

ref

where H is the incident field and H is the reflected field by the ground plane. H s c a t is the magnetic field radiated by the magnetic current source on St in the presence of the ground plane. On invoking the image theory, jjscat c a n k e written as H scat ( r ) =

_2jkoY0

ff

G(r.r') • M(r')dS'

(30)

J J Sot,

where S ^ denotes the infinite plane that coincides with the ground plane and G denotes the dyadic Green's function for free space. Since M has a spacial periodic property, it can be expanded in terms of Fourier series oo

M(x,y)=

Mpqe-j{k**x+ky>y)

]T

(31)

p,q= — oo

where Mpq = -LIx-ly

//

M(x,y)

e^k***+k^dxdy

(32)

J JSt

where kxp

=

k

TTrP "I" *xi J-x 2TT

("")

v
-2jk0Y0

Y.

G(kxp,kyq)-Mpqe-j^x+kyy)

(35)

p,q= — oo

where G is the spectral version of dyadic Green's function and it can be expressed explicitly as 2

kn "*U

G ( f c x p , Kyq) —

2

2jkzk0

— ^xp k

—k

k

r^xp^yq

2

—k ^xp^yq k 2 k,-. — k 2 "'O

^yq

(36)

Finite Element Modeling of Periodic

Structures

139

where "•z

=

—

y«0

"-xp

—

Kyq •

V"' /

The integral equation (35) relates the surface magnetic current to its magnetic field. It enforces the radiation condition on the top surface of the unit cell in the presence of an infinite array. It is therefore referred to as the periodic radiation boundary condition. By substituting (35) into (28) and then (5), the functional now becomes F(E) = i [[J

—(V x E) • (V x E) - fcgerE • E dV

+jk0Z0 HI E • JdV -2*jj//MJ J St

-2jk0Z0

II

&(kxp,kyq)-Mpqe-j{k*ox+kv^dS

J2 p, <7 =-oo

M • HincdS

+ other surface integral terms. (38) The discretization of the surface integral term (28) yields the corresponding matrix entries oo

&ij = —2,KQ±xly

y

JNj

• Ca-(/cXp, Kyq) • JN •

(39)

eil-k^x+ky"v)dS

(40)

Pi<7= — oo

where N f = —!— {I Ni TXTy J JSt and the RHS known vector entries hi = -2jkQZ0 J J

Ni • HincdS.

(41)

Numerically, the infinite summation in (39) is always truncated with —M < p < M and —N < q < N, where M and N are the maximum Floquet mode included in the summation. A sufficient number of Floquet modes needs to be included to obtain convergent result. One the other hand, the field expansion in the FEM determines that the variation in the FEM solution cannot exceed the maximum variation that the FEM mesh can resolve, which in turn depends on the mesh density and the element order. This fact imposes a limitation that M and N cannot be arbitrarily large.

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Fortunately, numerical experiments show that the magnitude of Floquet harmonics decreases rapidly with increasing p and q and an inclusion of seven to nine lowest Floquet harmonics is typically sufficient to produce convergent results. Since irregular elements are used, the evaluation of the integral in (40) requires numerical methods such as the Gaussian-Legendre quadrature. When St is large, the evaluation of the integrals may take a long time. Assembling of (39) leads to a linear system that is nonsymmetric and partially full, partially sparse. Such linear systems can be factorized and solved efficiently using the UMFPACK package. To show that the periodic boundary condition and the periodic radiation boundary condition correctly model the field behavior, a simple example is tested. The test example consists of a 20-cm-thick uniform dielectric layer composed of lossy material (e r = 3 — j) and backed with a ground plane. A plane wave is incident from above with incident angle 6 = 60°. The TE and TM power reflection coefficients, shown in Fig. 4, are calculated over 0.1 ~ 0.7 GHz frequency band. In the FEM calculation, the unit cell is modeled as a 20 x 20 x 20 cm homogeneous dielectric box with a ground plane placed on its bottom surface. The good agreement between the FEM results and the analytical solutions validates the proper enforcement of the periodic boundary condition and the periodic radiation boundary condition.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

frequency (GHz) Fig. 4. Power reflection coefficients for the 20-cm-thick uniform layer (e r = 3— j) backed with a ground plane at 9 = 60°.

Finite Element Modeling of Periodic Structures

141

In active array structures, such as array antennas, waveguides are often used to feed each antenna element. The waveguide considered here is a homogenous waveguide with an arbitrary cross section. For hollow waveguides, propagating modes are TE and TM modes. For coaxial lines, an additional TEM mode exists. Without losing generality, the electric field on the waveguide aperture Sw can be written as a sum of distinct waveguide modes: oo

E(r) = E

inc

Cm ero(r)e>'fc"»*

(r) + £

(42)

m=0

where

V^o - klm

kz

(43)

where kcm and em are cut-off wavenumbers and transverse mode functions associated with the mth mode. They can be obtained numerically by solving the corresponding 2-D eigenvalue problem using the FEM. The mode index m includes all possible modes (TE, TM, and TEM if possible), and E i n c is the incident field. For simplicity, we assume fundamental mode incidence; that is, E l n c = eo- The Cm is the reflection coefficient for each mode. Using orthogonality between different modes, Cm can be expressed as Cm = e~jk""z

ff e m • (E - Emc)dS. (44) JJsw A boundary condition can be derived by substituting (44) into (42), and then applying n x V x operator on both sides of the equation, yielding n x V x E - jk0Z0 ^

Ymem

\\

e m • EdS = -2jk0Z0Y0eo

(45)

where Ym are modal admittances given by h for TE modes k0Z0 Ym=

{

-fco kzmZo

for TM modes

— Z0

for T E M mode.

(46)

Applying boundary condition (45) on Sw yields an additional surface integral term in functional (5): 5(E) = ^

JJ

E

^

Y

mem JJs

e m • EdS J - 2Y0E • e0dS. (47)

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After discretization and field expansion, (47) can be cast into the following matrix format as Sij = jk0Zo J2 YmN?N™

(48)

771

bi = 2jk0Z0Y0N?

(49)

N? = ff

(50)

where N 4 • emdS.

Assembling the above matrix entries results in a full submatrix in the system matrix [K\, as in the case of the periodic radiation boundary condition. The evaluation of (48) requires the evaluation of the integral (50), which is also calculated numerically. In computer implementation, the infinite sum in (47) must be truncated. The number of waveguide modes included in (47) should be large enough to represent the actual field distribution on the waveguide aperture. But again, the variation of the highest mode included should not exceed the variation that the FEM mesh can resolve. To validate the waveguide port condition, a waveguide junction discontinuity problem is considered. The waveguide junction consists of two sections of concentric circular waveguides, with a radius of 1.5 m and 0.95 m, respectively. A TEu wave is incident from the larger section and the power reflection coefficient for the TEu mode observed at the larger section is calculated. In this example, the waveguide port conditions are imposed on the waveguide cross sections to truncate the computation domain, and 25 higher modes plus the dominant mode are included in the boundary condition. Figure 5 shows the calculated reflection coefficient, which agrees well with the solution obtained by the mode matching method. The numerical example shows clearly that the waveguide port condition correctly models the field behavior on the interface of the waveguide junction.

2.3. Fast Sweep

Techniques

The formulation presented in the previous two subsections is a frequencydomain method, which means that the system matrix has to be generated and solved for each specific frequency. In many practical applications, broadband frequency responses need to be calculated. One can imagine that the total time required for such frequency sweeps will become undesirably

Finite Element Modeling of Periodic

80

R,=1.5m

Structures

100

120

143

140

160

180

200

frequency (MHz) (a)

(b) Fig. 5. (a) Geometry of a circular waveguide junction, (b) Power reflection coefficient for TEn mode.

long. In this subsection, the AWE method is combined with the FEM formulation to perform fast sweeps. The AWE technique was first introduced in circuits analysis [12]. Its applications in the frequency-domain FEM and MoM have been introduced recently [13]-[15]. In the context of electromagnetic problems, the linear system of equations to be solved is [A] {x} = {6}

(51)

where both the matrix and the vectors are functions of wavenumber (or frequency). First we want to expand {x} in terms of a Taylor series (52) where fco is the expansion wavenumber. In order to calculate {x}^\ i = 1,2, • • • ,n, one can take the nth derivative of the both sides of (51) to give {[A] {x})in)

= {b}in)-

(53)

The left-hand side of the above equation can be easily expanded as

E 7i\(n 7 7—^ 7i)\M ( 0 {s} (n -° = {b){n)-

(54)

i=0

From (54), a recursive formula can be derived to calculate {x}^n\ given by n! •[A]M{x} (i)tT\(n-i) ^—' i\(n — i)

{*}<"> = [A]"1 {&}<»>-£

which is

(55)

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144

Jin

Once the Taylor coefficients are obtained, the Taylor series is then converted to the corresponding Pade series {x} =

TUaUk-koY

(56)

i + Y,?=i{bUk-ko¥ The Pade expansion is more capable of capturing the poles of the transfer function and thus yields a larger convergence radius. In cases where the convergence range of one single expansion point is not enough to provide an accurate solution over the entire frequency band, a multiple point expansion technique called complex frequency hopping (CFH) can be employed [13]. In CFH, a binary search algorithm is used to determine the expansion points adaptively until the entire frequency band is covered by the total convergence areas of the expansion points. The algorithm is outlined as follows. Given a sweep range [/cm;n, fcmax] and an error tolerance e: (1) Apply AWE at km-ln and fcmax; obtain the corresponding expansion functions x\(k) and X2(fc). (2) Let fcmid = (fcmin + fcmax)/2 and calculate xi(kmid) and x2(fcmid)(3) If \x\{kmii) — :c2(fcmid)| < e, the algorithm stops; otherwise, the above steps are repeated for subregions [fcmin, &mid] and [fcm;d,fcmax]. In general, the fast sweep technique described above is much more efficient in generating broadband frequency responses compared to the pointby-point approach. Figure 6 shows the reflection coefficient for a 1-D periodic dielectric structure over frequency band 130 ~ 176 MHz. The AWE solver gives the same result as the direct calculation at each frequency point. Table 1 shows the comparison of computation time for using the direct and AWE solvers. It is seen from the table that the total computation time required by the AWE solver is much less than that required by the direct solver. The convergence radius of an expansion point is mainly determined by two factors: the highest order of the derivatives and the accuracy of the derivatives. Because of the enforcement of the periodic boundary condition and the periodic radiation boundary condition, the derivatives of matrix entries with respect to wavenumber generally cannot be calculated analytically in closed forms, which is different from [16] where analytical derivatives are obtainable. Therefore, most derivatives are calculated numerically. This imposes limitations on both the highest order and the accuracy of the derivatives.

Finite Element Modeling of Periodic

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140

150

145

160

170

frequency (MHz)

(a)

(b)

Fig. 6. (a) Geometry of the dielectric layer with periodically varying permittivity with e r i = 1-44, e r 2 = 2.56, d = 1 m, h = 2.037 m, 9 — 45°. (b) Specular power reflection coefficient. Table 1. Solver type Direct AWE

Comparison of computation time for frequency sweep. Time for a single frequency point (s) 53.2 116.7

Number of points

Total time (s)

461 6

24525 700.2

Many important parameters of array structures, such as scan pattern, are angular responses. It is easily seen that the system matrix entries are dependent on incident angles (or scan angles) as a result of the enforcement of the periodic boundary condition (23) and the periodic radiation condition (35). This is different from the nonperiodic case, in which only the RHS known vector is angle-dependent and the system matrix is factorized only once for all angles of incidence. However, for periodic structures, the system matrix must be regenerated and refactorized at each incident angle. To accelerate wide-ranged angular sweeps, the AWE technique described above for frequency sweep can be formulated for angular sweep in a very similar fashion. In most scan analyses, 6 is the scan angle. It is therefore convenient to define an angle-dependent wavenumber kt = fcncosfl. Then every quantity in (51) can be written as a function of kt and its derivatives with respect to kt can be calculated. Following the same procedures as described in this section, one will finally obtain a set of expansion coefficients at several expansion points ktl, kt2, • • • ,hn, from which the desired quantity over the entire angular range can be extrapolated. Figure 7 shows the angular sweep of the specular reflection coefficient

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for the same example as shown in Fig. 6 at / = 140 MHz. The two curves calcuated by the AWE solver and the direct solver superimpose upon each other. Table 2 shows the comparison of computation time for calculating the entire angular response. It is evident that the AWE solver outperforms the direct solver in terms of total computation time. 1 0.9 0.8

efl

c n II / •= u a>

aCD

0.6 0.5

S

C U)

04

ro

z> 0.3 0.2 0.1

0

0

10

20

30

40

50

60

70

80

90

9 (degrees) Fig. 7. Specular magnitude reflection coefficient versus incident angle for the dielectric layer in Fig. 6 at / = 140 MHz.

Table 2. Solver type Direct AWE

Comparison of computation time for angular sweep. Time for a single frequency point (s) 53.2 116.7

Number of points

Total time (s)

90 8

4788 933.6

3. Numerical Results This section presents examples of FEM analysis of periodic structures including periodic absorbers, frequency selective structures, and phased array antennas. The numerical examples validate the FEM formulation outlined in Section 2 and demonstrate its capabilities.

Finite Element Modeling of Periodic

3.1. Periodic

Structures

147

Absorbers

Periodic absorbers are extensively used as low-reflection covering of the metallic shielding walls in electromagnetic compatibility measurements. Scattering analysis of periodic absorbers initially resorted to analytical methods. At very high frequencies, asymptotic methods such as the uniform theory of diffraction (UTD) [17] can be applied. At very low frequencies, the original problem can be transformed into a multilayer problem and approximate solutions can be obtained [18], [19]. Recent advances in full-wave numerical methods enable a more general and more accurate analysis. Periodic version of the MoM has been formulated and widely used [20]-[23]. There also exists frequency- and time-domain finite difference modeling of periodic absorbing structures [24]-[26]. The FEM is most suitable for modeling complicated absorber geometry and composition and has been utilized in the analysis and design of periodic absorbers [5], [27]. Figure 8 shows the unit cell in a periodic absorber array. The absorbing material is typically shaped into wedges for 2-D arrays and pyramid-cones for 3-D arrays. In the FEM analysis, the unit cell contains only one single absorber. On the four side surfaces of the unit cell, the periodic boundary conditions (23) are imposed. The top surface St is placed somewhere above the tip of the absorber and the periodic radiation boundary condition (35) is imposed there. The bottom surface Sb is placed on the physical boundary of the absorber, which is usually a PEC ground plane that serves as an electromagnetic shield. After the electric field is solved everywhere inside the unit cell, the specular reflection coefficient R (p = 0,q = 0) can be calculated from the aperture fields on the top surface St, giving

Hf f +Hf a t Hinc

= 1 - 2jfc 0 y 0 6(fc x0 , ky0) • MQO

(57)

where Y0 = 1/ZQ and M p q and G(kxp,kyq) are defined in (32) and (36), respectively. First, we investigate the absorption characteristics of a metal-backed planar stratified absorber, which has been analyzed with the surface integral equation method [22]. The absorber is made up of three infinite layers, each with a distinct material property. The geometry is shown in the inset of Fig. 9. For comparison, the material properties of the three layers have been chosen to be similar to those in [22]. The calculated specular reflection coefficients from 2 to 18 GHz are plotted in Fig. 9. For this problem, the

Z. Lou & J.-M. Jin Absorbing material

Ground plane Fig. 8. Unit cell in a periodic absorber array. analytical solution is obtained by cascading the propagator matrix for each layer. It is observed t h a t the numerical result agrees well with the analytical solution.

3.2mm

E

r1

3.2mm

6

r2

3.2mm

E

r3

mi nm mm) on -15--

—— Analytical O FEM -25

2

4

6

8

10

12

14

16

18

frequency (GHz)

Fig. 9. Specular power reflection for a stratified absorber at normal incidence. To evaluate the performances of higher-order vector elements, we reconsider the stratified absorber example by evaluating the root-mean-square

Finite Element Modeling of Periodic

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149

(RMS) errors for using different orders. To compare different orders, two different mesh sizes are used and for each of them the RMS errors are calculated using zeroth-, first-, and second-order tetrahedral elements, respectively. The RMS errors versus the number of unknowns are shown in Fig. 10. In the calculation, the frequency is fixed to be 30 GHz and the reflection coefficients are calculated versus incident angles. The RMS error is defined as 1

RMS \

N

°

(58)

i=l

where Ra and Ri are the analytical and calculated reflection coefficients, respectively, and Ns is the number of incident angles. It is obvious from Fig. 10 that the accuracy improves with increasing element order. It is also noted that, for the same number of unknowns, more accurate results can be obtained by using higher-order elements. This example demonstrates the advantage of using higher-order FEM. 10' - a - hfl.=0.13 - 0 - h/A.=0.1 10"

a> 1 0 CO

2nd

10

\ 10'

Fig. 10.

10° Unknowns (N)

2nd 10'

Comparison of RMS errors for different mesh sizes and element orders.

Second, we examine the reflection from a wedge absorber versus incident angles. The absorber is singly periodic in the x direction and uniform in the y direction, and has a relative permittivity er = 1.45 — 0.4j. The detailed geometry is shown in the inset of Fig. 11(a). The plane of incidence

Z. Lou & J.-M.

150

Jin

is assumed to be the xz plane. Figures 11(a) and (b) compare the results by the FEM code and those by a low-frequency approximation method (or the so-called homogenization method) for / = 100 and 300 MHz, respectively. In the low frequency approximation, the absorber is replaced with several homogeneous layers, whose effective permittivity and permeability are determined from the profile and material composition of the original absorber. The approximation is valid as long as the period of the absorber is small compared to the wavelength. The comparisons show that at lower frequencies the two methods predict a very similar reflection pattern, while at higher frequencies some discrepancies are observed. In the latter case, the period is slightly larger than one half of the wavelength. In this example, the absorber is not backed by a conducting plane.

5I 0

i 20

i 40

i 60

i 80

-45' ' 0 10

I 100

' 20

e (degrees) (a)

' 30

' 40

' 50

' 60

' 70

' 80

' 90

6 (degrees) (b)

Fig. 11. Specular power reflection for the wedge absorber with e r = 1.45 — 0.4j. (a) / = 100 MHz. (b) / = 300 MHz.

Next, we present some examples of doubly periodic absorbers. Figure 12(b) shows the specular power reflection coefficient for a doubly periodic absorber array whose unit cell geometry is shown in the inset. In this example, the permittivity of the absorbing material is a function of frequency, as shown in Fig. 12(a). To implement the fast frequency sweep, it is more convenient to expand the permittivity as a set of known functions of frequency, such as polynomial functions

e(/) = E « „ f l - - ^ - ) n n=0

V

(59)

/max/

where an are the expansion coefficients determined from the material's per-

Finite Element Modeling of Periodic

Structures

151

mittivity characteristics and / m a x is the maximum sweeping frequency. Typically, N = 4 is sufficient for a good modeling of realistic permittivities. The specular reflection coefficient is calculated for frequencies between 30 and 200 MHz. Results from three different methods are compared: the FEM, the finite-difference frequency-domain (FDFD) method [24], and the lowfrequency approximation method. It is observed that the three methods agree very well with each other for lower frequencies and differ by about 2 dB at higher frequencies. As another example, Fig. 13(b) shows the geometry and calculated specular reflection for a doubly periodic absorber of similar pyramid shape. The frequency-dependent permittivity used in this calculation is given in Fig. 13(a). The FEM result is compared with the low-frequency approximation and with the results by Jiang and Martin [5] for the 50 MHz to 1 GHz frequency range. Again at lower frequencies, the three results agree better. The differences at higher frequencies may due to the low power level of the reflected field (less than —50 dB). Also the condition for the low-frequency approximation to be valid may not be satisfied at these high frequencies.

J

•

o FDTD FEM — Low frequency approximation

'\/*^^fei-._

•5; 0.15

0.1

frequency (GHz)

(a)

0.1

0.15

frequency (GHz)

(b)

Fig. 12. (a) Complex permittivity and (b) specular power reflection for a metal-backed pyramid absorber with period of 24 in and height of 96 in at normal incidence.

Finally, we investigate the effect of the pyramid profile on the absorber performance. Figure 14 shows three different pyramid profiles that are under consideration. They all have the same height and base dimensions, but the side surfaces of the pyramid are flat, concave, and convex, respectively. The radius of curvature R is 70 in for both concave and convex surfaces. The calculated specular reflection is shown in Fig. 15. The results show that,

Z. Lou & J.-M.

152

Jin

o Jiang et al. — FEM — Low Frequency 0.4

0.6

0.2

0.8

frequency (GHz)

0.4

0.6

0.8

frequency (GHz)

(a)

(b) Fig. 13. (a) Complex permittivity and (b) specular power reflection for a metal-backed pyramid absorber with period of 8 in and height of 24 in at normal incidence. R=70

Fig. 14.

Side view of pyramids with flat, concave, and convex side surfaces.

except at very low frequencies, the flat-surface pyramid outperforms both the the concave- and the convex-surface pyramids. 3.2. Frequency Selective

Structures

Frequency selective structures (FSSs) are 2-D arrays of metal and/or dielectric configurations, whose scattering response is characterized by a passband or stopband, at which total transmission or reflection occurs. FSSs have been analyzed using different methods. Analytical study of 1-D periodic dielectric layers was made in [28] based on a guided wave theory. Concerning numerical methods, integral equation based methods, such as the MoM, can be employed [29]-[31]. Time-domain analysis of FSSs was reported recently using the FDTD method [32]. Since the FEM can easily model irregular shapes, it is often preferred for more complicated structures. Successful FEM analysis of FSSs has been reported in [4] and [33]. In addition, a hybrid method that combines the FEM with the general-

Finite Element Modeling of Periodic Structures

153

0

-10

-20

2T n

-40

-50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

frequency (GHz) Fig. 15. Specular power reflection for metal-backed pyramid absorbers with flat, concave, and convex side surfaces at normal incidence.

ized scattering matrix (GSM) method through a cascading procedure was proposed to maximize efficiency [34]. Figure 16 shows a generic FSS unit cell. It may consist of one or several dielectric layers which contain metal surfaces or structures. For simplicity, the metal surfaces are considered infinitesimally thin and perfectly conductive, but this is not required in the formulation. In fact, because of the volumetric discretization, the metal and dielectric can have any shape and composition. The unit cell is enclosed by four side surfaces and a top and a bottom surface. On the four side surfaces, the periodic boundary conditions (23) are imposed. On the top and bottom surfaces, the periodic radiation boundary conditions (35) are imposed with the incident term in (35) excluded for the bottom surface Sb- After the electric field is solved for everywhere inside the unit cell, the reflection coefficient can be calculated using (64) and the transmission coefficient can be calculated similarly as T =

Hscat t Hinc

= -2jk0Y0C.{kM,

kyo) • Moo

(60)

where Moo is calculated from (32) with the surface integration carried out on SbFirst, we consider a 1-D periodic dielectric layer whose permittivity

154

Z. Lou & J.-M.

Jin

Unit Cell

Conducting Surfaces Fig. 16.

Front view of a generic frequency selective structure.

alternates between e r i = 1.44 and er2 = 2.56 along the x direction, as illustrated in Fig. 6(a). A TE plane wave is incident from one side of the slab at the incident angle 6 = 45°. Two cases are calculated, with normalized slab height h/d chosen to be 1.713 and 2.037, as specified in [28]. The computed reflection coefficients are shown in Figs. 17(a) and 17(b). For comparison, the analytical solutions by Bertoni et al. [28] are also plotted. It is shown that the FEM accurately predicts the analytical resonant bands in both cases. We also note that only five to six expansion points are used in the fast frequency sweep, but all the detailed features have been captured. r-e

o

\ ' i\

FEM Bertoni

y \

V

<

| o.4h o 0.

e o

fi

R

fifl

71

Normalized wavenumber k„h (a)

2.5

3

3.5

4

4.5

5

5.5

Normalized wavenumber l^h (b)

Fig. 17. (a) Specular magnitude reflection coefficient for h/d = 1.713. (b) Specular power reflection coefficient for h/d = 2.037.

The next example concerns a circular slot FSS whose unit cell geometry

Finite Element Modeling of Periodic

Structures

155

is shown in Fig. 18(a). Figures 18(b) and 18(c) show the power transmission coefficients at normal incidence for the free-standing FSS and the FSS printed on a 0.3-/im-thick dielectric substrate with permittivity e r = 16, respectively. Both results are compared with those by Volakis et al. [4]. When a coarser mesh is used in the calculation of Fig. 18(c), the peak of our result shifts toward a longer wavelength.

(c)

Fig. 18. (a) Top view of a circular slot FSS. (b) Specular power transmission coefficient for the free-standing FSS. (c) Specular power transmission coefficient for the FSS with substrate.

The next example involves an embedded FSS structure. The unit cell is a rectangular patch laid between two 1-mm-thick dielectric layers with permittivity e r = 2.0, which is shown as the inset of Fig. 19. The power reflection coefficient for normal incidence is shown in Fig. 19. The result agrees well with the MoM result by Mittra et al. [29].

156

Z. Lou & J.-M.

Jin

frequency (GHz) Fig. 19.

Specular power reflection coefficient for the embedded patch FSS.

Finally, we present a comparison between numerical results and measured data. The structure of this example is illustrated in Fig. 20(a). Two rectangular microstrip patches reside on the top and bottom surfaces of a uniform dielectric slab with permittivity e r = 2.5. A PEC screen is placed in the middle of the slab and a rectangular slot is cut on the screen so that the two patches are coupled. The transmission coefficient for this FSS structure has been measured using the waveguide simulation measurement technique in [31]. The comparison between the FEM result and the measured data is shown in Fig. 20(b). In the calculation, the incident angle varies according to the measurement setup described in [31].

3.3. Phased Array

Antennas

Phased array antennas are of great importance in modern radar and communication systems. Recent integrated antenna technology has brought large arrays into widespread applications, and a good understanding of such arrays generally requires numerical analysis. Accurate prediction of array parameters using numerical methods not only reduces development costs and design time but also renders invaluable information to the design engineers. Available numerical methods include the MoM, the FEM, and some hybrid methods [3], [7], [8], [35], [36]. Among these methods, the MoM has been most extensively employed for numerical analysis, and verification has been made by comparing with experiment results for various

Finite Element Modeling of Periodic

Structures

157

/ V •

/A

FEM — Measured

,y

y,

£ 0-6

/•My'

- p

(0

7>


"5 0.4 " • i f •••

.. i\\>.r

VN

/ •

V 34.04mm

(a)

3

3.25

3.5

Frequency (GHz)

(b)

Fig. 20. (a) Geometry of the aperture-coupled patch FSS. Dielectric rilling is not shown in the figure, (b) Calculated and measured specular magnitude transmission coefficients.

antenna configurations. Increasing demands on array performances often require complicated designs with inhomogeneous material composition and irregularly shaped conducting patches. To accurately model the electromagnetic field behavior in each array element and the mutual coupling between the elements, a 3-D full wave analysis is necessary. As another aspect of numerical modeling, an accurate analysis of radiation problems is impossible without a careful modeling of the feeding structures, which itself can be complicated for a successful design. In view of these reasons, the FEM appears to be most suitable as an analysis tool due to its general formulation and versatility in geometry modeling. This subsection describes the use of the FEM for a 3-D radiation analysis of infinite array antennas. The FEM discretization is confined to a single unit cell whose side surfaces and top surface are imposed with appropriate periodic boundary conditions. The coaxial line feed is modeled in an accurate manner using the FEM mesh. On the coaxial port aperture, a waveguide port condition is imposed to truncate the mesh. The numerical results obtained from this precise coaxial modeling are compared with the results obtained using the short current probe feed. For wide-band calculations, the AWE technique is used to accelerate frequency and angular sweeps. Numerical examples are given for microstrip-patch arrays and flared notch antenna arrays.

Z, Lou & J.-M. Jin

158 3.3.1. Feed Modeling

An important numerical issue in the analysis of radiation problems is the proper modeling of the feeding structures. For the MoM, a precise modeling of the detailed feeding structure is difficult, due to the 3-D nature of the feeding structures. In [35], an idealized current probe was used to model the coaxial feed. As shown in Fig. 21(a), the simplified probe-feed model consists an idealized current filament with an infinitesimal radius and a short electrical length. A constant current amplitude and phase are assumed on the filament. Although the simplified model predicts fairly accurate results for microstrip array antennas with a thin substrate and reveals very useful information on scan characteristics, it does not take into account the effects of the probe radius and the variation of the current along the probe, and thus is not applicable to microstrip-patch arrays with electrically thick substrates. On the other hand, accurate feed modeling can be easily realized using the FEM approaches. By applying a very fine mesh to the feed region, one can accurately model the detailed geometries of the feeding structures. For a coaxial feed, a conducting probe is modeled as the extension of the central conductor of the coaxial line, as shown in Fig. 21(b), and the PEC boundary condition is imposed on all the conducting surfaces. On the coaxial aperture, the waveguide port condition given in Subsection 2.5 is imposed. Using the above approach, we are able to include the effects of the probe radius as well as the current variation along the probe. The rapid field variation around the coaxial-substrate junction region is also captured.

(a)

(b)

Fig. 21. Illustration of (a) short current probe feed and (b) coaxial line feed. Both simplified and accurate feed models can be easily implemented in a FEM code. For the simplified model, the source current distribution J s is assumed to be Js = I0S{x0,y0)

0
where IQ is the current on the probe which is positioned at (xo,yo)

(61) an

d

Finite Element Modeling of Periodic

extends from the ground plane obtained by solving the system as 1 Zin = -j2I o

Structures

159

up to a height h. Once the electric field is matrix, the input impedance is calculated fh / E- zdz Jo

= -^E(/

IVidzW

(62)

where i accounts for all the unknowns that are associated with the probe. For the accurate coaxial model, the reflection back to the coaxial line is calculated as the product of the reflected field with the dominant waveguide mode

"11.

e 0 • (E -

Emc)dS

s,„

= Y/N?Ei-l

(63)

i

where Sw denotes the coaxial line aperture and i accounts for all the unknowns on Sw. 3.3.2. Numerical Examples First, we present several microstrip-patch array examples for the purpose of validation since this type of array has been thoroughly analyzed in the literature. Figure 22 shows the general geometry of a microstrip patch array, in which metal patches are placed periodically on a grounded dielectric substrate with height h and permittivity e r . The patches are fed with coaxial lines that enter the substrate from below the ground plane. In order to model an infinite patch array on a substrate, the unit cell is usually properly chosen so that it contains only one patch and the periodic boundary condition is imposed on its four side surfaces. After the fields are obtained everywhere inside the unit cell, desired parameters such as input impedance, active reflection coefficient, and active element pattern can be easily computed. The appropriate definition for the active reflection coefficient is given by

_ zin(0,4>)-zin(o,o) -zin{o,) + zLM-

m
(64)

The use of the conjugate operator in (64) discounts the imaginary part of the input impedance and more clearly reveals the scan properties of the array.

Z. Lou & J.-M.

160

Jin

W

Fig. 22.

Infinite microstrip patch array on substrate.

Pozar and Schaubert [35] presented a detailed MoM analysis of microstrip patch arrays which is used here as reference. The example considered is a 0.3Ao x 0.3Ao rectangular patch array with periodic length Tx = Ty = 0.5Ao- The permittivity of the substrate is eT = 2.55. E-plane active reflection coefficients are calculated for substrate height h = 0.02Ao and O.O6A0. The results calculated using the simplified and accurate coaxial feed models are plotted in Fig. 23 together with Pozar's MoM result, which used a simplified probe feed. It is noted that all three results agree with each other very well. All of them predict a scan blindness at 6 = 68.8° for h = O.O6A0, as a result of surface-wave excitation. Moreover, the agreement between the simplified and precise feed models suggests that scan performances such as active reflection coefficients are insensitive to the specific feed scheme, as indicated in [35]. The next example is a microstrip patch array consisting of 0.25Ao x 0.25Ao rectangular patches with periodic length Tx = Tv = 0.5Ao and er = 2.5. Figure 24 shows the comparison between the input impedances calculated with the simplified probe feed and accurate coaxial line feed for two different substrate thicknesses. It is noted that the input resistances calculated by the two models agree better when the substrate is thinner (h = 0.05Ao). In the case of the thicker substrate (h = O.lAo), the difference between the two models is more pronounced, although the resonant phenomenon becomes less obvious. In both cases, the accurate coaxial model predicts a slightly lower resonant frequency than does the simplified probe model. Next, we consider a circular patch array fed with coaxial lines. The periodic lengths are given by Tx = 34 mm and Ty — 36.1 mm, and the patch

Finite Element Modeling of Periodic

Structures

161

Pazar, probe model • FEM, probe model FEM, coaxial model

Pozar, probe model • FEM, probe model FEM, coaxial model

I 0.4!

40

60

40

6 (degrees)

60

e (degrees)

(a)

(b)

Fig. 23. Active reflection coefficients for the microstrip patch array consisting of 0.3Arj x 0.3A0 rectangular patches, Tx=Ty = 0.5A0, tr = 2.55. (a) h = 0.02A0, s = 0.075A0; (b) h = 0.06A0, s = 0.14A0.

-*-•••*•• ••«••

Coaxial, real part Probe, real part Coaxial, imaginary part Probe, imaginary part

g-200

-o

' •

Coaxial, real part Probe, real part Coaxial, imaginary part Probe, imaginary part

20

*»i*A±*iiiii>* -20 -40

2

2.5

3

2

2.5

frequency (GHz)

frequency (GHz) (a)

(b)

Fig. 24. Active input impedances for the microstrip patch array consisting of 0.25Ao x 0.25A0 rectangular patches, Tx = Ty = 0.5A0, s = 0.0625A0, e r = 2.5. (a) h = 0.05A; (b) fe = 0.1A.

radius is RQ = 14.29 mm. The outer and inner radii of the coaxial line used in the simulation are r0 — 1.492 mm and r^ = 0.456 mm, respectively. The S11 parameter for the coaxial port is shown in Fig. 25. Both the magnitude and phase of the calculated £ u parameter agree with the results obtained by the hybrid generalized scattering matrix (GSM) and the FEM [8]. In the FEM calculation, the incident angle varies according to the waveguide simulator measurement in [8]. The flared notch array antenna, or the Vivaldi array antenna, is wellknown as a good candidate for wide-band and wide-scan antennas. Figure 26

162

Z. Lou & J.-M.

Jin

•

-^0.6

FEM » FEM/GSM

frequency (GHz)

(a)

frequency (GHz)

(b)

Fig. 25. (a) Magnitude and (b) phase of the Su parameter for the microstrip patch array consisting of circular patches with radius RQ = 14.29 mm, h — 0.79 mm, e r = 2.33. Feed is modeled using a precise coaxial line model with r0 = 1.492 mm, ri = 0.456 mm. Feed offset is s — 5 mm, and permittivity of the coaxial line is ec — 2.024.

shows the unit cell geometry of a flared notch array antenna. A conducting patch is printed on one side of a dielectric slab, which stands vertically above a ground plane. A narrow slot line is cut in the middle of the conducting patch and is gradually flared into an open mouth to provide a smooth impedance transition from the feeding network to the free space.

Metallization

Dielectric Slab

Coaxial Aperture Fig. 26.

Ground Plane Unit cell of the flared notch antenna array.

The antenna protype and parameters used in our calculation follow a design described in [7]. The antenna is configured for single-polarized operations and is fed by a short section of a coplanar waveguide that connects the balanced coaxial line and the unbalanced slotline. The coplanar waveguide

Finite Element Modeling of Periodic

Structures

163

consists of two arms: one evolves as the flared slotline and the other ends at a hollow circle which serves as a wide-band open circuit. The coplanar waveguide is then connected to a coaxial line by connecting their central conductors. The advantage of this design is that metallization is applied to only one side of the dielectric slab and no via-holes are needed for ground connections. The array spacing in the x and y directions is Tx and Ty, respectively. The height and thickness of the dielectric slab are 33.3 and 1.27 mm. The half width of the slotline varies with z according to an exponential function given by w{z) = 0.25e0-1232.

(65)

This function gives a half width of 15 mm at the open mouth. The hollow circle has a radius of 2.5 mm. The coaxial line is modeled as an aperture cut on the ground plane with its inner and outer radius being 0.375 and 0.875 mm, respectively. Figure 27 shows the active reflection coefficient as a function of frequency at broadside scan angle. The three curves correspond to Tx = 32,36, and 40 mm, respectively. In all these cases, Ty is fixed to be 34 mm. It is found that a sharp reflection peak appears in all three curves and the corresponding frequency of the peak increases as the periodic length in the x direction is reduced. A similar behavior has been observed previously in the scan performances of single- and dual-polarized tapered slot antenna arrays with double-printed substrate, which is referred to as impedance anomalies [37]. These impedance anomalies have been found to relate to the resonances of the cavity formed by the substrate region. Here we employ the same cavity model, with the 2-D cavity defined by the two adjacent rows of radiating patches and the ground plane. By assuming the PEC conditions on the two side walls and the bottom surface and the PMC condition on the top surface, one can easily calculate the resonant frequencies of the 2-D cavity. The dominant mode is the TEu mode and the corresponding resonant frequency is given by

where Tx is the periodic length in the x direction and h is the height of the substrate. For the three different values of Tx in Fig. 27, the calculated resonance frequencies are 4.30, 4.73, and 5.19 GHz, which are close to the reflection peaks (4.24, 4.6, and 5.08 GHz) in Fig. 27. It is also observed that the resonance frequency does not vary when the scan direction is steered

Z. Lou & J.-M.

164

Jin

off broadside, further indicating that the phenomenon is associated with certain geometrical parameters of the unit cell itself. — i —

0.9

- -

0.8 c

0.7

efl

g

t3

0.6

Tx=32 mm Tx=36 mm Tx=40 mm

-

" -

J

CD

'i

on -a

0.5

L-. V.

0.4

" \ \:

3

V\ v.

CO

2

/^T<-'r''r" ""~t"-^.

\v-

\\" \ >

0.3

\ *• -

0.1

-

•

^

J

V-

0.2

/

//-•

f

-

2.5

3

3.5

4

4.5

5.5

frequency (GHz) Fig. 27. Active reflection coefficient for the flared notch antenna array with different array spacing in the x direction.

Next, the scan performances of the flared notch array are studied. Figure 28 shows the E-plane and H-plane active reflection coefficients at / = 2.5, 3.0, and 4.8 GHz. The figure indicates that input impedance changes smoothly with scan angle over a wide angular range at both 2.5 and 3.0 GHz. At 4.8 GHz, where the array spacing becomes larger than onehalf wavelength, the grating effect-induced scan blindness appears in both E-plane (6 = 57°) and H-plane (9 = 47°) scans. In the E-plane, there is an additional blindness at 9 = 17°, which agrees with the result by McGrath [7]. Some researchers have explained this phenomenon as the excitation of certain waveguide modes on a corrugated structure [38]. 4. Conclusions This chapter describes an FEM analysis of infinitely periodic structures. The theoretical basis for the analysis is the fundamental FEM theory combined with the Floquet theorem and the integral equation formulation. Higher-order vector basis functions are used to expand electromagnetic fields. Numerical results demonstrate the efficiency and accuracy of using higher order vector basis functions. The AWE-based technique is combined

Finite Element Modeling of Periodic Structures

f=2.5 GHz - - - f=3.0 GHz f=4.8GHz

f=2.5 GHz - - - M.O GHz f=4.8GHz
/ ' ' /

to

5 ,

20

165

40

8 (degrees) (a)

jT

0.6

a.

,'

20

40

6 (degrees)

(b)

Fig. 28. Active reflection coefficient for the flared notch antenna array with Tx = 36 mm at / = 2.5, 3.0, and 4.6 GHz. (a) E-plane scan, (b) H-plane scan. with F E M t o perform fast frequency and angular sweeps. T h e F E M feed modeling is also discussed in detail for radiation problems. T h e validity and versatility of t h e F E M formulation have been demonstrated t h r o u g h numerous numerical examples presented in Section 2. Most of the examples are presented for validation purposes, through which comparison with previously published results can be made. More sophisticated designs are not covered in this chapter, but they can be analyzed easily by the same m e t h o d developed here due to the flexible volumetric modeling of the F E M .

Acknowledgments This work was supported by a grant from O N R under contract number N00014-01-1-0210. References 1. S. Gedney and R. Mittra, "Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution," IEEE Trans. Antennas Propagat, vol. 39, pp. 1605-1614, Nov. 1991. 2. J. M. Jin and J. L. Volakis, "A hybrid finite element method for scattering and radiation by microstrip patch antennas and arrays residing in a cavity," IEEE Trans. Antennas Propagat, vol. 39, pp. 1598-1604, Nov. 1991. 3. E. W. Lucas and T. P. Fontana, "A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays," IEEE Trans. Antennas Propagat., vol. 43, pp. 145153, Feb. 1995. 4. J. L. Volakis, T. F. Eibert, D. S. Filipovic, Y. E. Erdemli, and E. Topsakal,

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6.

7.

8.

9. 10.

11. 12.

13.

14.

15.

16.

17.

18.

Z. Lou & J.-M. Jin "Hybrid finite element methods for array and fss analysis using multiresolution elements and fast integral techniques," Electromagn., vol. 22, pp. 297-313, May-Jun 2002. Y. Jiang and A. Q. Martin, "The design of microwave absorbers with highorder hybrid finite element methods," 1999 IEEE Antennas Propagat. Soc. Int. Symp., vol. 4, pp. 2622-2625, 1999. J. M. Jin and J. L. Volakis, "Scattering and radiation analysis of threedimensional cavity array via a hybrid finite-element method," IEEE Trans. Antennas Propagat, vol. 41, pp. 1580-1586, Nov. 1993. D. T. McGrath and V. P. Pyati, "Phased array antenna analysis with the hybrid finite element method," IEEE Trans. Antennas Propagat, vol. 42, pp. 1625-1630, Dec. 1994. M. A. Gonzalez, J. A. Encinar, J. Zapata, and M. Lambea, "Full-wave analysis of cavity-backed and probe-fed microstrip patch arrays by a hybrid modematching generalized scattering matrix and finite-element method," IEEE Trans. Antennas Propagat, vol. 46, pp. 1998, Feb. 1998. J. M. Jin, The Finite Element Method in Electromagnetics, Wiley, New York, 2nd ed., 2002. R. D. Graglia, D. R. Wilton, and A. F. Peterson, "Higher order interpolatory vector bases for computational electromagnetics," IEEE Trans. Antennas Propagat, vol. 45, pp. 329-340, Mar. 1997. I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices, Clarendon Press, Oxford, 1986. L. T. Pillage and R. A. Rohrer, "Asymptotic waveform evaluation for timing analysis," IEEE Trans. Comput.-Aided Des., vol. CAD-9, pp. 352-366, April 1990. Q. J. Zhang, M. A. Kolbehdari, M. S. Nakhla, and R. Achar, "Simultaneous time and frequency domain solutions of em problems using finite element and cfh techniques," IEEE Trans. Microwave Theory Tech., vol. 44, pp. 15261534, Nov. 1993. M. Li, Q. J. Zhang, and M. S. Nakhla, "Finite difference solution of em fields by asymptotic waveform techniques," IEE Proc. H, vol. 143, pp. 512-520, Dec. 1996. C. J. Reddy, M. D. Deshpande, C. R. Cockrell, and F. B. Beck, "Fast RCS computation over a frequency band using method of moments in conjunction with asymptotic waveform evaluation technique," IEEE Trans. Antennas Propagat, vol. 46, pp. 1229-1233, Aug. 1998. J. Liu, J. M. Jin, E. K. N. Yung, and R. S. Chen, "A fast three-dimensional higher-order finite element analysis of microwave waveguide devices," Microwave Opt Tech. Lett, vol. 32, pp. 344-352, Mar. 2002. R. T. Tiberio, G. Pelosi, and G. Manara, "A uniform gtd formulation for the diffraction by a wedge with impedance faces," IEEE Trans. Antennas Propagat, vol. 33, pp. 867-873, Aug. 1985. E. F. Kuester and C. L. Holloway, "A low-frequency model for wedge or pyramid absorber arrays—I: theory," IEEE Trans. Electromagn. Compat., vol. 36, pp. 300-306, Nov. 1994.

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19. E. F. Kuester and C. L. Holloway, "A low-frequency model for wedge or pyramid absorber arrays—II: computed and measured results," IEEE Trans. Electromagn. Compat., vol. 36, pp. 307-313, Nov. 1994. 20. R. Janaswamy, "Oblique scattering from lossy periodic surfaces with application to anechoic chamber absorbers," IEEE Trans. Antennas Propagat., vol. 40, pp. 162-169, Feb. 1992. 21. N. Marly, D. D. Zutter, and H. Pues, "A surface integral equation approach to the scattering and absorption of doubly periodic lossy structures," IEEE Trans. Electromagn. Compat, vol. 36, pp. 14-22, Feb. 1994. 22. N. Marly, D. D. Zutter, and H. Pues, "Integral equation modeling of the scattering and absorption of multilayered doubly-periodic lossy structures," IEEE Trans. Antennas Propagat, vol. 43, pp. 1281-1287, Nov. 1995. 23. C. F. Yang, W. D. Burnside, and R. C. Rudduck, "A doubly periodic moment method solution for the analysis and design of an absorber covered wall," IEEE Trans. Antennas Propagat, vol. 41, pp. 600-609, May 1993. 24. W. Sun, K. Liu, and C. A. Balanis, "Analysis of single and doubly periodic absorbers by frequency-domain finite-difference method," IEEE Trans. Antennas Propagat., vol. 44, pp. 798-805, June 1996. 25. K. Liu, W. Sun, and C. A. Balanis, "A periodic finite-difference method for microwave absorber analysis and design," 1994 IEEE Antennas Propagat. Soc. Int. Symp., vol. 3, pp. 1400-1403, 1994. 26. C. L. Holloway, P. M. McKenna, R. A. Dalke, R. A. Perala, and C. L. Devor, "Time-domain modeling, characterization, and measurements of anechoic and semi-anechoic electromagnetic test chambers," IEEE Trans. Electromagn. Compat, vol. 44, pp. 102-118, Feb. 2002. 27. Y. C. Chung, B. W. Kim, and D. C. Park, "Range of validity of transmission line approximations for design of electromagnetic absorbers," IEEE Trans. Magn., vol. 33, pp. 1188-1192, July 2000. 28. H. L. Bertoni, L.-H. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propagat, vol. 37, pp. 78-83, Jan. 1989. 29. R. Mittra, C. H. Chan, and T. Cwik, "Techniques for analyzing frequency selective surfaces—a review," Proc. IEEE, vol. 76, pp. 1593-1610, Dec. 1988. 30. J. M. Jin and J. L. Volakis, "Electromagnetic scattering by a perfectly conducting patch array on a dielectric slab," IEEE Trans. Antennas Propagat., vol. 38, pp. 556-563, Apr. 1990. 31. R. Pous and D. M. Pozar, "A frequency-selective surface using aperturecoupled microstrip patches," IEEE Trans. Antennas Propagat, vol. 33, pp. 653-678, May 1998. 32. J. Vazquez, C. G. Parini, and P. J. B. Clarricoats, "The finite difference modelling of infinite arrays in time domain," National Conference on Antenna and Propagation, pp. 209-212, Aug. 1999. 33. I. Bardi, R. Remski, D. Perry, and Z. Cendes, "Plane wave scattering from frequency-selective surfaces by the finite-element method," IEEE Trans. Magn., vol. 38, pp. 641-644, Mar. 2002. 34. M. Lambea, M. A. Gonzalez, J. A. Encinar, and J. Zapata, "Analysis of fre-

168

35.

36.

37.

38.

Z. Lou & J.-M. Jin quency selective surfaces with arbitrarily shaped apertures by finite element method and generalized scattering matrix," 1995 IEEE Antennas Propagat. Soc. Int. Symp., vol. 3, pp. 1644-1647, June 1995. D. M. Pozar and D. H. Schaubert, "Analysis of an infinite array of rectangular microstrip patches with idealized probe feeds," IEEE Trans. Antennas Propagat., vol. 32, pp. 1101-1107, Oct. 1984. C.-C. Liu, A. Hessel, and J. Shmoys, "Performance of probe-fed microstrippatch element phased arrays," IEEE Trans. Antennas Propagat., vol. 36, pp. 1501-1509, Nov. 1988. T.-H. Chio H. Holter and D. H. Schaubert, "Elimination of impedance anomalies in single- and dual-polarized endfire tapered slot phased arrays," IEEE Trans. Antennas Propagat, vol. 48, pp. 122-124, Feb 2000. D. H. Schaubert and J. A. Aas, "An explanation of some e-plane scan blindnesses in single-polarized tapered slot arrays," IEEE APS Int. Symp. Dig., vol. 33, pp. 1612-1615, June 1993.

FACTORIZATION OF POTENTIAL A N D FIELD DISTRIBUTIONS WITHOUT UTILIZING THE ADDITION THEOREM Ali-Reza Baghai-Wadji Vienna University of Technology Institute for Industrial Electronics and Materials Science Gusshausstr. 27-29, A-1040, Wien Vienna, Austria, EU E-mail: alireza. baghai-wadji@tuwien. ac. at ErPing Li Institute for High Performance Computing Computational Electro-magnetics & Electronics 1 Science Park Road, #01 — 01 The Capricorn Singapore Science Park II Singapore 117528 E-mail: [email protected] The classical TV—body problem consists of analyzing the state of a system of N interacting bodies, where the force exerted on each body in the system results from its interaction with all the other bodies in the system. There are typically 0(N ) operations necessary to compute the forces involved. In solving Singular Surface Integrals (Boundary Element Method, Method of Moments), we encounter a similar problem: Discretizing the underlying surface integrals, and, thus, introducing N unknowns, we generally need to compute a large number of interactions requiring 0(N2) operations. Thereby, the interaction coefficients depend on the relative Euclidean distance between the interacting elements in a rather complicated form. Hierarchical tree-structured algorithms have been developed to reduce the complexity of computations. The key to the majority of standard fast algorithms is the availability of potential functions in factorized form, which are obtained by applying the so-called Addition Theorem. A question arises as how to proceed if a factorized form is not available, or cannot be easily constructed. It is the prime purpose of this work to present a general recipe for factorizing potentialand field distribution functions without utilizing the Addition Theorem.

169

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A.-R.

Baghai-Wadji

& E. Li

1. Introduction The construction of multipole expansions for potential- and field distribution functions relies on the existence of certain factorization formulae, which are conventionally formulated by using the so-called Addition Theorem, [l]-[5]. The main objective in this contribution is the presentation of a recipe for obtaining factorized forms without the need for the Addition Theorem. Factorization, as used here, refers to the multiplicative decomposition (tensor product) of the potential functions into functions, which depend on either the co-ordinates of the source- or the observation point. The multiplicative decomposition is crucial in all those instances, where we need to, e.g. integrate over a large number of source points, while keeping the position of observation point invariant. Such circumstances routinely arise in large scale engineering computations, which are based on singular surface integrals, [6]-[10]. Plainly speaking, the basic idea behind the factorization is, to break down the Euclidean distance between the source- and the observation point rso = |r s — r 0 | into terms, which depend on the co-ordinates of the source point position vector r s and the observation point position vector r 0 , separately. There are several possibilities to this end, which we briefly outline in the sequel. • Standard multipole expansions and the underlying Addition Theorems result in factorized forms which involve functions in real space. • Inhomogeneous Expansions: An alternative approach for obtaining factorized forms consists of employing integral transforms with certain translation-invariant properties; a prominent example is the Fourier transform, which results in inhomogeneous spectral decompositions of the form

with 3t{7(fci,fc2)} < 0, [11]-[12]. At this point several comments seem to be in place. (i) As an example, assume 7(^1, fo) = — (kf + k\)1^2 = — \k\, and f(ki,k2) = 1/|A;|. (The two dimensional inhomogeneous spectral domain decomposition of the fundamental solution of Poisson equation in three dimensions leads to an equation, which involves such 7 and /.) Integrating with respect to k\ and ft^i both varying from

Factorization

of Potential and Field

Distributions

—oo to oo, we obtain 1/R with R being the Euclidean distance between two points with the position vectors r = (zi, 22,0:3) and r =

(Xi,X2,x$):

R2 = On - x[)2 + (x2 - x'2f + (x3 - 4 ) 2 Therefore, while in spectral domain we have the factorization of the terms ePkl(~Xl~x'^, ejk^X2~x'2\ and e-Mx3±x'3\^ t a k e n t h e s p e ctral constituents collectively, the factorization gets lost; we obtain l/R, which is no longer factorized. (ii) Homogeneous Expansions: A similar statement can be made if we consider equivalent homogeneous expansions of the form: f{kljk2,k3)ejkl{xi~x'l)ejk2ix2-x^ejk:'{x3-x^ Integrating with respect to k\, k2, and £3, which vary from —00 to 00, the factorization loses its morphology. (iii) Assume TV source points pi (i = 1, • • • ,N) and one observation point q. Employ for each {pi,q)—pair an "appropriately" chosen local co-ordinate system, with the corresponding spectral decomposition of the above-mentioned inhomogeneous exponential type. It can be shown, that the introduction of local co-ordinate systems is instrumental in achieving fast algorithms based on factorization. As an example consider the Fast-MoM, which is a variation of the Method of Moments, [8], [9], devised to accelerate the numerical computations, by separating the topological and metric properties of problems, [13]. The Fast-MoM introduces and utilizes certain types of problem-independent "universal functions," which derive from the factorization based on inhomogeneous spectraldecomposition, [14]. (iv) It should be pointed out that obtaining factorized forms in spectral domain is based on the diagonalization of the underlying partial differential equations with respect to a distinguished direction in space, [15]-[17]. In the course of our discussion the reader will be familiarized with some of these notions and results. (v) In this paper we will diagonalize the Poisson equation with respect to the radial direction and use field expansions in real space. Diagonalization will guide us to the factorization of the field distribution functions without utilizing the Addition Theorem. (vi) The diagonalization of PDEs offers a second type of factorization, which is partially in spatial domain, and partially in spectral

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Baghai-Wadji

& E. Li

domain. To be more specific, assume we have diagonalized the governing equations with respect to the z—co-ordinate. Let the fields be decomposed in the form g(x,y,x',y')f(z,z'). If the condition g(x,y,x',y') = h(x — x',y — y') is satisfied, we can decompose g into its spectral components, each having the form: g(ki,k2)ejklixi-x'^ejk2ix2-x^ Obviously integrating with respect to k\ and k2, both varying from —oo t o oo, we obtain g(x — x',y — y'). This means that while the function g(x — x',y — y') in the (x, y)— plane is not factorized, its spectral components g(ki, k2)e:>hl<'Xl~Xl',e:>k2(X2~x^ are. T h e content of the remaining part of the paper is organized as follows: We first will briefly discuss the Addition Theorem as used in the electrostatic problems. Then we show how to diagonalize the Poisson's equation in spherical co-ordinates, by introducing a constructive recipe. Finally, we provide a detailed discussion on how to factorize the field expansion functions. The derivation procedure should be appealing to engineers and professionals in various disciplines, who will easily recognize the wide range applicability of the proposed concept. It can be shown that the proposed recipe is applicable to problems in electrodynamics and elastodynamics, involving isotropic, anisotropic, or bi-anisotropic materials. Furthermore, it can be shown t h a t whenever a system of PDEs permits diagonalization, our procedure is applicable. In a recent contribution we formulated a conjecture claiming t h a t every physically-realizable linear system of PDEs allows diagonalization, with reference to a certain distinguished direction in space, [18]. Based upon this conjecture it is reasonable to expect, that the application of our recipe to various physical problems, will result in novel factorized expansion formulae, without utilizing the Addition Theorem.

1.1. Addition

Theorem

for Spherical

Harmonics

Consider the position vectors r i and r being specified by the spherical coordinates (ri,9i,<j>i) and (r,#,<£), respectively. Let the angle between r i and r be 7 (Figure 1). It is known that the electric potential measured at r due to a unit point charge located at r^ read, [1]: i

°°

;

Pi{cos

T^rr^^ I1

ll

<

i=o

r

^

>

(1)

Factorization of Potential and Field Distributions

173

Fig. 1. The (r,9,(f>)— spherical co-ordinate system. In the figure are shown the unit point charge qi with the position vector r i , and the co-ordinates (ri,9i,tf>i), and the observation point specified by the position vector r and the co-ordinates (r, 6,4>). The angle between the two position vectors is 7. The relative position of the observation point to the source point is denoted by the vector r s o . Here, r< = inf{r, r\) and r> = sup{r, r i } . Furthermore, P; (•) stands for the Legendre polynomial of order I, which has the following representation known as the Rodrigues' formula:

It is straightforward to show that the Legendre polynomials form a complete set functions on the interval — 1 < x < 1. It is also known t h a t COS7 can be expressed in the following form: COS7 = cos#cos#i + sin#sin#icos (4> — 4>i) • The Addition Theorem expresses P; (C0S7) in terms of the spherical harmonics Y ; m (#i, <j>i), and Y; m (9, dS), depending on the angles 6\, 4>i and 6,
A.-R. Baghai-Wadji & E. Li

174

4 Pi (cos7) = 2 ^ 1

' E

Y

?m ^ , 0i) W ,

0)

(3)

7 7 1 = —Z

The proof of this relationship relies on several definitions and equations, which we have summarized below for the sake of completeness [1]. Expression for the Spherical Harmonics Ylm{e^)

=

^ ^

F r i c o s e ) e

^

(4)

me H,-(z-i),---,o,---,(/-i),z], with PJ™ (•) being the associated Legendre polynomial. For positive m, the associated Legendre polynomial is defined by the expression jm

P|"(x) = ( - l ) m ( l - z 2 )™/ 2 —P,(o;).

(5)

Furthermore, it can be shown that the following relationship holds true

PTm(x) = ( - l ^ l ^ j l p j " ^ ) .

(6)

P,m(-) and e j m * are eigenfunctions of certain commuting operators, which we do not discuss here in further detail. We merely wish to point out the relevance of the commutativity property of operators: Given £ = £i£2 with £i£2 = £2£i, then the eigenfunctions of £, say, /(•), is the tensor product /i(-)/2(0 with /i(-) and /2O) being the eigenfunctions of the operators £1 and £2, respectively. Further fundamental relationships are the normalization and orthogonality conditions

J dOsmO J d0Yf,m, (0, 0) Yim (0,0) = 5Vi6m,m, 0

(7)

0

and the completeness relation 00

E (=0

1

£ m=-l

Y

lm (A'- *') Ylm (*, 0) = S (cOS0 - COS0') 8 (0 - 0') .

(8)

Factorization

of Potential and Field

Distributions

175

In (7) Ski is the Kronecker delta symbol with the definition (1

k =l

if

*« = {o

(9)

ootherwise. th—

In (8) 6(-) is the Dirac's delta function, which is a distribution [10]. Substituting (3) into (1) we obtain 1

oo

I

I

i

4

YLie

i r ^ r ^ £ 2fTi^ 1=0 m=—l

'^')YimieA)-

(10)

>

This is the celebrated factorized form for the electric potential measured at the point r due to a point charge located at the point r' in terms of their spherical co-ordinates (r,6,<j>) and (r',61, cf>'), respectively. The significance of this factorization becomes evident in computations, e.g. the boundary element method, where we need to integrate over a large number of source points, while keeping the observation point unchanged. The main purpose of this paper is to present a general procedure for the factorization of field distributions in terms of functions, which individually depend on the co-ordinates of the observation- and source points, without utilizing the Addition Theorem. 2. Field Expansions in Real Space In this section we outline a technique for obtaining field expansions for electrostatic fields without utilizing any Addition Theorem. We illustrate the underlying key ideas in terms of the following elementary problem: Consider N isolated point charges of strengths qi (i = 1, • • • , N). Let the location of i t h point charge be specified by the position vector r;, or, alternatively (rj, 0i,(j>i) in a spherical co-ordinate system. It is an elementary task to obtain the electric potential distribution tp(r, 6,
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Theorems. The main objective in this work is to show that we alternatively can achieve this goal without utilizing any Addition Theorem. We provide a simple recipe for factorizing potential functions. The proposed factorization technique is quite general and applies to problems for which no factorizations have been known so far, e.g. the factorization of EM fields in anisotropic or even bianisotropic media. In this paper, however, we limit ourselves to electrostatic problems in free space. Our recipe involves the following steps: (1) Diagonalize the governing equations with respect to the radial coordinate r. (2) Identify the "interface conditions," which automatically arise in the diagonalized equation. (3) Consider the spheres r = T{ which partition the entire space into N +1 nested spherical layers: r < r\, 7"j_i < r < r* (i = 2, • • • ,N), and rjy < r. (4) Use the following solution ansatzes for the potential functions if^ (r, 6, <j>) and the radial component of the dielectric displacement

In the first layer: oo

¥> ( 1 ) M,0) = £

/

£

[A\y]Ylm(e,d>)

(11a)

1=0 m=-l

^'(r.M) = *—' E E - r KM 1WM) £ L J 0

(Hb)

1=0 m=—l

In ith layer with i = 2, • • • ,N: OO

I

(i)

¥> (r,M) = £ £

{ [A«r']yJm(M)

(=0 m=-l

\nlmrl+l

"+" OO

1

I

(12a)

j

^-2>WM,0) = £ £ {- i 0

Ylm(9,<j>)}

[A^T1]

y,m(M)

i=0 m=-J +

l+ l r

1 ' ri+i

Ylm{9,)}

(12b)

Factorization

In (N + l)th

of Potential and Field Distributions

177

layer:

V>("+1V, «,*) = £ £

B,lm

ie')M,^EE i+i

r'+l

Ylm(6,4>)

(JV+l)

B,lm

pi+1

(13a)

y;m(^,
Considering the general expressions given in (12), it is evident that for obtaining the series expansions for the first layer we need to set the coefficients B equal to zero; likewise to obtain the series expansions for the N + 1 layer we have to set the coefficients A equal to zero. (5) Identify the interface conditions from the diagonalization of the Poisson equation. (6) Satisfy the interface conditions, with the following point source functions:

Q^(r,6,<j>) = q~6(r

- n)6{cos6 - cos0i)6((l> -

fa)

(14)

(7) Utilize the orthogonality relationships for the spherical functions H m ( M ) , Eq. (7). (8) Determine the unknown coefficients [^J„T-'] and [-Bj^/r'l}]. It is worth mentioning that the coefficients [jBim /r1^1] result in the standard field series expansion for r > rjv, used in Fast Multipole Method, we would like to emphasize that the proposed procedure provides not only expansion formulae for r > rjv, as is the case in conventional procedures, but also the series expansions for the remaining layers, i.e., in entire space. It is also worth mentioning that we will obtain recurrence formulae for the determination of the coefficients [Aj^rJ] and

3. Diagonalization of Poisson's Equation in Radial Direction in Spherical Co-ordinates Our starting point are the Maxwell's equations (15a) and (15c) in the electrostatic limit and the constitutive equation (15b) in free space:

A.-R.

Baghai-Wadji

& E. Li

V•D = p

(15a)

D = £0E

(15b)

E = -grad<^

(15c)

with

p(r, 9, <j>) = pi— S (r - n) 6 (cosO - cos#;) 6{<j>-

fa)

(16)

representing a point charge with the intensity pi, and being located at the point (ri,6i,(pi). Here, D is the dielectric displacement vector, E is the electric field,

D = -eogradip

(17)

Our goal is to diagonalize the system of equations summarized in (18) with respect to the radial variable r:

V • D = Pi-?6 (r - n) 6 (cos0 - cos^) <$(<£D = -eogrady

fc)

(18a) (18b)

We observe that the radial position n subdivides the entire space into the subspaces 0 < r < n and n < r. Introducing the Heavisides' unit step function H(£) according to

(19)

we can write the vector field D and the scalar field if in the following form:

Factorization

of Potential

and Field

Distributions

179

D (r, 9,4) = D> (r, 6, <j>)H{r- n) + D< (r, 6, <j>) H (n - r)

(20a)

V? (r, 0, <j>) = v* (r, 6, <j>)H{r- r4) + <^< (r, 0,0) tf (r4 - r)

(20b)

Substituting (20a) into (18a) results in V D = V • [D > if (r - n) + T>) H(r-

(21a)

n) + D^-grad [H (r - n)}

+ ( V - D < ) F ( r i - r ) + D < grad[H ( r — r ) ] = pi -zS (r - r») 6 (cosO - cos0,) 5(4>- 4>A .

(21b) (21c)

Likewise, substituting (20b) into (18b) gives

D> H (r - n) + D
(22a)

>

= - e 0 (grad

(22b)

In spherical co-ordinates the grad of the scalar function f(r:6,
grad/=fer

+

ige 9 + - ^ | e ,

(23)

Applying (23) to the Heaviside's function H(r — ri) and H(ri — r) results

grad [H(r - n)] = ^ — T i 2 e

= 5(r - r,)e r

(24a)

grad [H(ri - r)} = -Sin - r)er = -6(r - ri)er

(24b)

r

Substituting (24) into (21c) and (22b) leads to

A.-R.

Baghai-Wadji

(V • D>) H{r-

n) + (V • D<) H fa - r)

>

+D 6(ri - r)e r - D<6(r = Pi—S(r-ri)6

& E. Li

n)er

(cosO - cos6i) <5 (0 — 0»)

(25)

and

G>H(r-ri)

+

D
= - e 0 (grad

) H (r - n) - E0 (grad(/?<) H (n - r) = -eo^Sfr

<

- r{)er + e0ip 5{r - n)er,

(26a) (26b)

respectively. With D > e r = D> and D < e r = Df and the fact that the supports of the functions H (r — r,), H (r* — r), and S(r — rt) are mutually exclusive, and, at the same time, their union spans the entire r—axis, we can split (25) and (26b) into the following equations: (V-D>)H(r-ri)=0

(27a)

(V-D<)ff(ri-r)=0

(27b)

D> (r, 6,4>) 6(r - n) - D< (r, 6, <j>) 6(r -

n)

= pi-^6 (r - n) 6(cos0 - cos6»i) 6 ( -
(27c)

D> (r, 0,0) H (r - n) = -e0 ( g r a d ^ (r, 0,0)) H {r - n)

(28a)

D< (r, 0,0) # (r4 - r) = - e 0 (grad^< (r, 0,0)) if (n - r)

(28b)

and

- e o ^ (r, 0,0) <5(r - r,)e r + e 0 y < (r, 0, ) <5(r - ri)e r = 0, respectively. It is worth mentioning that integrating

(28c)

Factorization of Potential and Field Distributions

181

(27c) with respect to r from 0 to oo results in the "interface condition" for D r at r = rj. Likewise, multiplying (28c) from both sides by e r and integrating the resulting equation with respect to r from 0 to oo, we obtain the "interface condition" for ip at r = r,. Putting together what we have found so far, we obtain the equations in (29), (30) and (31), which are, respectively, associated with H(r — n ) , H (n — r) and S (r-j — r):

H{r-n):=*{

(29)

H(ri-r):=*{

(30)

[D>(rue,<j>)-D<(ri,e^) — Pih& ( cos # _ COS#i) 0~{4>~ 4>i) 6 (u - r) :-

(31) ^>(ri,6,)-)

=0

Since in the domain r < Ti and r > Ti we have the same differential equations (Eqs. (29) and (29)) the distinction conveyed through the signs < and -* becomes obsolete, and therefore, they will be omitted subsequently. With this omission our governing equations are: V•D = 0 D = -eogrady?

(32a) (32b)

The divergence operator V- applied to the vector field F in spherical co-ordinates is defined as follows:

' • » - ? 5 (•"•')+ ^ ( - «

+sb^

P3)

182

A.-R.

Baghai-Wadji

& E. Li

We earlier had denned the grad operator, Eq. (23). With these definitions (32) can be written in the form:

_

dip

D =

1 dip

.„ , ,

-£°-&7e>- ~ £°-r-deee ~ £o7^6d^^

I dip

(34b)

Writing the vector equation (34b) in the form of three scalar equation (34) gives the following set of first order equations for the four scalar functions ip, Dr, Dg, and D$.

?|(^) + sbl< s '" s "'» + sb^= 0

(35a)

Dr = - e o ^

(35b)

Dg = -e0-%s

(35c)

D

* = -£°7to%

^

It is a simple exercise to convince ourselves, that upon substituting (35b), (35c), and (35d) into (35a) we obtain the standard Laplace equation for the potential function
Factorization of Potential and Field Distributions

r2

r

gr \

I

i

£o

d ( sme

r^edG(

id? £o

rde

183

1 dy N = 0 rsvaO d(j) \ rsinO d(j>, (36a)

-1DP £o

T

= *£

(36b)

dr

In the second and third terms in (36a) the term 1/r can be moved behind the d/d6 and d/d

i d £o sin0 06

(37a)

dr

£o r" q2

(37b)

H^-B^-S^)

Note t h a t in (37a) we have introduced r2Dr. have


f n 7*2

£

. o s b s 5 (sin<%) + £ 0 ^ 3 5 *

Written in matrix form we

r2D.

°

_9_ dr

We can obtain an alternative formulation by considering the relationship

!L(r2Dr)=2rDr

+ r2^rDT

(39)

in (38), i.e.,

£0

£o?r r2sin9

08

(sm6w)

+

e0ris]n'ieg^

2 r

" 'P ' .Dr.

d '

Obviously considering tp and DT (instead of ip and r2Dr) as essential field variables destroys the antisymmetric structure of the matrix at the LHS of (38).

A.-R.

184

Baghai-Wadji

& E. Li

We conclude this section by the following comment, which is significant in numerical calculations. Introducing the variables y/£o

\/eoV

iib^( s i n 6 l M)

+

^e3^

0

y/£0
d_ dr r2-Sa=

(41)

4. Derivation of Green's Functions in Spherical Co-ordinates In what follows we apply the above mentioned recipe to the simplest possible case, i.e., the determination of the potential distribution due to a point charge source in free space. In an (r, 9, cp) spherical co-ordinate system consider a point charge specified by the strength qi and the coordinates (ri,9i,<j>i). As mentioned earlier, the radial distance r\ of the source point from the center of the co-ordinate system subdivides the entire space into two parts: r < r\ (domain I) and r > r\ (domain II), which are separated by the sphere r = r\. In the following the field variables in the domains I and II, respectively, will be denoted by the superindices 1 and 2.

4.1. Potential

Distribution

For the potential distributions associated with q\ we obtain the following expressions in the regions I and II: oo

I

^)(r,M) = E E 4&y*WM) /=0 m=—l oo I

2

(42a)

*

^ >M,4>) = E E Bft^YUO,®

(42b)

1=0 m=-l

The choice of these expressions for the potential function guarantees that the series in (42) are finite in their respective domains of definitions.

Factorization of Potential and Field Distributions

185

Domain II

Fig. 2. Given a single point charge q\, we can select an arbitrary spherical coordinate system (r,6,). A sphere with its center at the origin of the chosen co-ordinate system, and passing through the position of the point charge divides the entire space into the domains I (0 < r < r\) and the domain II (r\ < r).

4.2. Radial Displacement

Components

Using the relationship Dr = —eod

oo

D^{r,e,cj>) = £ (=0 oo

D?\r,e,) = E

I

J2

{-eo^'-'JlWM)

(43a)

m=-l I

E

1=0 m=-l

f

1 1 ° %(1 + !)3+2 Yim(0,
e B

*•

(43b)

'

For quick reference we summarize t h e formulas in (42) and (43) in t h e following slightly modified form:

A.-R.

186

Baghai-Wadji

oo

I

^1)(r,6»,0) = £

£

1=0 oo

^)(r,M) =

& E. Li

(44a)

[^VjlWM)

m=-l

Z

£

£

(2) 1 B,im Ylm{0,) r;+i

(44b)

/=0 m = - / oo Z

^(r,«,0) =

5 : E { - e o ; [ ^ r ' ] } y l l B ( t f9,4) I

(44c)

(=0 m = - i oo £

Ylm{9,CJ>)

(44d)

As will be clear shortly, it is computationally advantages to work with the coefficients [-A^rj] and [BJ^/r1^1] as our unknowns, rather than A)^ and B J m . The square brackets in (44) are meant to point out this fact. For the determination of [-*4;mVi] and [-Bjm /r^"1"1], involved in these equations, we utilize the "interface conditions," which are formulated in the next section. 4.3. Interface

Conditions

The solutions in regions I and II satisfy the following interface conditions on the (ri)— sphere: (45a) •Dl1Hrt,0,) + Di2\rt,9,) = gi-^<5(cos6> - cos0iW(/> - 4>i)8{r - n) where we have introduced rt defined as

4.4. Determination

(45b)

lim {n ± el = r, .

of the Coefficients [A^r1^

and

For the determination of the coefficients [-Aj^r^] and [-BjJ/r[ +1 ] (I £ N°,rn = —/,••• ,1) we proceed as follows (N° denotes positive integers): (i) Substitute Eqs. (44) for the fields into the interface conditions (45).

Factorization

of Potential and Field

Distributions

187

(ii) Multiply both sides of the resulting equations by Yt*m, (9, <j>) (I € N°, m = —I, • • • ,1). Here and in the following * stands for the complex conjugation, (iii) Exchange the order of summations and integrations, (iv) Integrate the terms on both sides of the equations over the surface of the r\ — sphere. (Recall that the surface element on the r\— sphere is r\d9sm.9d(t>.) (v) Use the orthogonality relationship for the spherical functions (7). In the following we apply the steps listed in the above recipe to ), tpW(rf,9,i) individually. We calculate the projection of the terms in the interface conditions (45), separately. In this way we hope to convince the reader from the wide range applicability of the method, and the fact that the derivation can be automated using symbolic manipulations. 4.4.1. Projection of the Potential Function ) onto the Co-ordinate Function Yi' m '(0,0) lit

TV

J d9sin9 J d
^¥^,(9,)(rrf

0 7T

/- „

27T

= J d9sin9j # Q

E

Q

i

£

[AW('T)'] > W M )

V 1=0 m = — I

xy; m ,(0,<£)(rr) 2 OO

(46a)

I

= (T)2£ E KW] 1=0 m=-l

x J d9sin9 J <Wm,(0,4>)Yim(9, ) 0

(46b)

0 oo

2

/

5

= (-D E »' E l—Q oo

[MnlirlY] < W

(46c)

m=-l

= K ^ E - ^ K ^ r ) ' ]

(46d)

1=0

= (f)2

[4Urif]

(46e)

188

A.-R. Baghai-Wadji

& E. Li

In transition from (46b) to (46c) we used the orthogonality relationship (7).

4.4.2. Projection of the Potential Function ) TV

2n

J desine J dfoW (r+,o, <£)y,rm, (e, 4,) ( r + )2 0 2TT

00

f dOsinO / " # < J T

1

1

E B (2)

Ylm{QA)

1=0 m=-

xy;m,(0,0)(r+)2sin^#

(^)2E=0 m=-l £

B,

(47a)

(2)

2TT

J dOsmO J d4Ytrm, (9, )Y,m (9, <£) 0

{rt?

(47b)

0 ! S,?(2) i m ' '(r+)''+ 1 .

(47c)

4.4.3. Interface Condition for the Potential Function By virtue of (45a) we equate the results obtained from (46e) and (47c) and obtain

M)

X

/ _ - / _ o(2)

(48)

Replacing V and m' by I and m, respectively, together with setting rf = r]*" = ri we obtain: 4(1)-J _ p ( a )

2

(49)

Factorization

of Potential and Field

Distributions

189

4.4.4. Projection of the Radial Displacement Function Dr ' (rx , 9, ) onto the Co-ordinate Function Yvm< (9, )

(9, <j>) (rf ) 2

J d9sm9 J dDP (r-,9,0)y;m, o o

= fd9sm6 / # J E E

{-^)\A^)l]Ylm{9,4>)

xy; m ,(0»(rf) 2

(50a)

= (OaE(-^)E[^(T)'] x /" d9sin9 J d<S>Y{,ml (9, )Ylm(9, <j))sm9d9d<j) o o = (n)2(-eo^)[A^UrT)1'}

(50b)

(50c)

4.4.5. Projection of the Radial Displacement Function D-.(2)^ r '(r1 ,9,<j>) onto the Co-ordinate Function Yi>m> (9,4>) ir

2n

J d6sm9 f dfiDW (r+,9, 0)y,?m, (9, cj>) (r+ o 2TT

oo

= /d9sin9 J'dJ£

/

Y, U^

xY,1„,(S,4,)(rt)2

(2)

J-

B' m (r+)'+! I W M ) (51a)

i\ ± r. 2x

x y dOBinff Jdd>Y!n.(9,

_ W)'(«£il) [B£

)Ylm(e, <£)

'

(51b) (51c)

A.-R.

190

Baghai-Wadji

& E. Li

4.4.6. Projection of the Source Term pr<5(cos# — cos6i)5{(j) — fa)6(r — r\) onto the Co-ordinate Function Yi'm>(6, fa 2-rr

oo

dr o

o o

x { gi *(cos0 - coS61)6((/> - fa)6(r -

roll^MXri)

2

}

7T

= qi I d6sin66(cos9 — cosOi) o f 2ir

\

oo

x J JdS(

l {r\) Jdr8{r

- n)i

=v,Tm,(».*i)

(52a) 7T

gi / d0sin0<5(cos0 - cos9{)Y{!ml (0, fa)

(52b)

«ir,r m ,(fli,0i)

(52c)

Substituting the results obtained in (50c), (51c), and (52c) into (45b) we have

(rn

a

(^)[^(rr)'']+(rf)

a

(

e

„^i

! I'm' ( r + ) J ' + l

D(2)

x=qiYlrm,(el,fa)

(53)

Using r1 = rf = r i , and replacing /' and m' by / and m, respectively, we arrive at:

i[AJJM]+(/ + !)

(2) £ Jm

1

J+l

eon

YLifiufa)

We can write (49) and (54) in the matrix form as follows:

(54)

Factorization of Potential and Field Distributions W l - 1 ' r AAlmrrl

"1

1

191

0 (55)

./

1

R «

1 + 1.

L

r

l

-1

.^r^(^^i).

Multiplying both sides of (55) by the matrix

1 21 + 1

7+ 1

1

-I

1

(56)

which is the inverse of the matrix at the left-hand side of (55) we obtain A(1)rl

1 2/ + 1

A

lmrl

R (2)

1

Z+ l

0

1

(57a) -I

1

^ m ^ l . ^ ) .

M™(01.*l) (57b)

21 + 1 LeSr^m(»i.*i)

Or, written explicitly,

(58a)

EQTX 21 + 1

[Blm^[+1

91

eon

(58b)

2T+-iY^e^-

•*. A ( 1 V

Vi

"*

A

lmri

Fig. 3. Network representation of the algorithm for the calculation of the expansion coefficients - A ^ n in the 1st layer.

Or, finally, in order to draw a conclusion, we write

192

A.-R. Baghai-Wadji & E. Li

,(2)J_ B lm 1+1

qi

V,

r

Fig. 4. Network representation of the algorithm for the calculation of the expansion coefficients M?j m Ai + * in the 2nd layer.

9i Y? WiM e o n 21 + 1 r[ m

(1)

B\


(2)

e 0 n 2/ + 1

(59a)

ri+117m(0i,0i).

(59b)

Substituting A\J and B\Jl into the expressions for (/^^(r, 9, 0) and ip{2){r,e,<j>) in (42) we obtain:

£STT;T 1 S - ( *-*'> oo

rlYlm{6,4>)

(60a)

/

^)(r-^,^=x:E

9i ! j+iv, e 0 r i 2/ + ^ Y ^ i ^ i )

1=0 m=-l 1 XYi {9,) .;+i m

(60b)

The potential functions ) and ip^(r,9,(p) constitute the Green's function associated with the point charge q\ located at (ri,0\, \). Note t h a t by construction (60a) is valid for r ). Furthermore, note t h a t (60b) is valid for r > r\. The condition r\/r < 1 guarantees the convergence of the series for ip^(r, 9, <j>). As is evident from the above derivation procedure the solution of the potential distribution associated with a point charge can be facilitated by recasting the original problem into a two-layer problem with the charge point being located at the interface of the two layers, on the r\ —sphere.

Factorization

of Potential and Field

Distributions

193

4.5. Potential Functions Due to two Point Charges on the Same Sphere Prom the above analysis it is immediate that for two point charges q\ and q2 being located at two points with the co-ordinates (r%, 6i,(j>i) and ( n , 62, ) in regions r r\, respectively:

L

1=0 m— — l

92

+ OO

e0ri 2/

(

-


(61b)

92

eon 21 + : We will return to these formulae when we have obtained the functions associated with three layers in the next section. 5. Three-Layer Problem In this section we consider two point charges q\ and q2, which we assume to be located at (ri,6i,<j>i) and (r2,62,(^2)'• The radii r\ and r2 (ri < r2) partition the entire space into three regions Region I: r < r\, Region II: r\ < r < r2, Region III: r2 < r. With the formula for the radial component of the displacment vector Dr(r,0,<j>), i.e., Dr(r,0,<j>) = —eodf/dr, we have the following expressions for the potential and the dielectric displacement in each region: Region I: 00

1

) = E E [A\y] Y^6^

(62a)

1=0 m=-l

DW(r, M) = E E ( -£) (62b) =0 m=-l

A.-R. Baghai-Wadji & E. Li

194

Domain III

Fig. 5. In the figure are shown two point charge sources, q\ and (72, with the coordinates {ri,0i,<j)i) and (7"2,02>02), respectively, with respect to an arbitrarily chosen co-ordinate system. In this case two spheres with their centers at the origin of the co-ordinate system, and radii r\ and T2 ( n < T2), passing through the position of the charges q\ and 92, respectively, subdivide the entire space into three domains: Domain I (0 < r < n ) , domain II (ri < r < T2), and domain III (7-2 < r).

00

I

^>M,4>) = £ £

{ [A&V]ylm(M)

(63a)

1=0 m=-l (2)

+

00

(

\Blm-[TT ,

,(M)}

. x

D?\rA4>)=Y, £ { (-*o;) [^V] y,m(M) + Uo

i+ 1

(2) 1 B,lm _/+l

ljm(M)}

(63b)

Factorization of Potential and Field Distributions

195

Fig. 6. In this figure it is shown that we may equally well choose the reference co-ordinate system (£,77, C), rather than the (x,y,z) co-ordinate system.

Region III: 8)

00

/

*< (r, »,*) = £ £ Z=0 00

R(3)JL" l+l

"Im

r

Ylm{6,cf>)

(64a)

B{3)

(64b)

m--l I

(•m

lm

— Ylm(6,<j>) rl+l

S o u r c e F u n c t i o n s The source functions are specified as follows:

Q^(r,

6,) = qAs(r

- nMcose

- c o s f c ) ^ - &)

i = 1,2 In Figure 6 it is shown that we may choose any reference co-ordinate system (£, rj, C) instead of the (x, y, z) co-ordinate system. In this way we

A.-R. Baghai-Wadji & E. Li

196

obtain various equivalent representations for the potential- and field distribution functions in entire space. A very important question for applications is the relationship between the expansion coefficients A and B in the system (x, y, z) and those in the system (£, rj, £)• This point will not be addressed in this paper. Solution of the Problem For the determination of the various coefficients A and B in (62)-(64) we proceed as follows: (1) (2) (3) (4)

Impose the "interface conditions" on the r\— and r 2 — spheres. Multiply both sides of each of the resulting equations by Y{tm,{6, <j>). Integrate over the surfaces of the r\— and r 2 — spheres. Utilize the orthogonality condition in (7). Applying these steps we obtain:

^^ri+flft'-^

(65a)

'4™M - '42M + (' + M22 4 T = ^-y,*ro(fli.0i)

(65b)

ry (2) l

(2)

Eo^i

( 3 )

A r + R — - B — lmr2 + &lm T+T _ °lm T+T

A

r

2

r

ffi^ (.0t)CJ

2

u\y2 ~(i + i)B% ^n + (i + i ) C } ^ r 92 " £or Y^(92,
(65d)

Comment: Consider the coefficients -A^r'J an( ^ - ^ m / r 2 + 1 L which are the field expansion coefficients in the first and the last regions in the present problem, in greater detail: The first region is fully specified by the radius ri (r < 7*1). The last region is specified by only one radius r 2 (r > r 2 ). The second region, or generally speaking, every other region i (i = 2, ••• ,N) is characterized by two radii ri-i and r». Coming back to the coefficients L ^ V J and LBj m /r2 +1 we recognize that in L4/mV' the superindex of A and the subindex of r coincide. Furthermore, we realize that in 5; m '/r 2 + 1 the superindex of B equals the subindex of r minus one. Therefore, whenever, the superindex of A and the index of the associated r coincide, or the superindex of B and the index of the associated r are equal we speak of the natural order of indices. If the natural order is violated

Factorization

of Potential and Field

Distributions

197

in a certain equation we can restore it by multiplying and dividing by a nondimensional quantity as explained in the following examples. Example 1: In (65a) the term Afjri satisfies our condition: The superindex A and the first index of r are equal. But what about the term ^ . ^ r ' ? Obviously our condition is violated. However, we can write l(2) Note that the "cor[4mM] = (»i/ri) [A\y2], by introducing 1 rection term" is smaller than unity: r\/r\ = ri/r2) < 1. Example 2: In (65a) the term [ J S ^ M * 1 satisfies our condition: The superindex B equals the subindex of r minus one. But what about the term 2?j m '/r 2 +1 in (65c)? Obviously our condition is violated. However, we can

i]-

write

[B£M

+ 1

]

- ( r i + 1 M + 1 ) [B^/r1^},

by introducing

[B%/r[^\ +1

Note that here also the "correction term" is smaller than unity: r j /r 2 + 1 = (ri/r2)l+l < 1. Performing these modifications, and dividing the second and forth equations by I + 1 (65) can be written in the following form:

,(2)_J_

["SM]-£) [ ^ H

lm

(66a)

l+l

(2)J_

B,lm

l+l

r

l

1i £0^1 I + 1

(66b)

ir m (0i,&) B{2)

— r

i

l+l

T+iKKl-(s) ^ 7r ^ 7 ^ ( ^ 2 , 0 2 ) £0 2 * + 1

i

m B (3) J+l

(66c)

'2

(2)_j_ B lm l+l

+

(3)_J_ Blm l + l (66d)

Note also that the division by I 4- 1 results in the term 1/(1 + 1) and 1/(1 + 1) which are both less than unity for I > 0. Writing in matrix form we have:

A.-R.

198

Baghai-Wadji

lm

["£M]-[(s)'

& E. Li

£•

(67a) (2) 1

B,lm ZJ+T

AWri

A

lmr2

+

iTi KM] [-*(*)'

(2) l1 r +i

B,lm

9i

(67b)

eori I + 1 YtnVuti)

4(2)^ J+i

(S)

j(2) 1 Jm -.'+'

(3)_J_ 5,lm J+l

(67c)

'9

4 (2 V

A -fef] 92

(2) DR

i^7TTy'»<*'*'>

1 lm ZTT

+

(3)_L_ J+l '2 -

B <™

(67d)

The above equations can be cast into the following matrix equation:

Factorization

of Potential and Field

-1

(-)'

i_ +1

l

Distributions

4™M

1

(T71

-(n.\l HI \r2J

0

1

A

\l+l

B(2)

1

5 (3)

1

-1 1+1

°

1

r

lm 1

L

I ( n. r2J

!

199

i+T - fe) ^iM^i^r)

(68)

L^TTT^mCa.^)

It is worth mentioning that the only information needed to construct the matrix at the left-hand side are the nondimensional quantities 1/(1 + 1) and (ri/r 2 ).

5.1.

Calculation

of the

Coefficients

While we can solve (68) straightforwardly it is instructive to eliminate the coefficients appearing in the boundary regions, i.e., in the first and the last regions first. The benefit of this elimination becomes apparent when we consider more general multilayered problems in the next section. Using (67a) and (67c) we can obviously eliminate L^JJ^i and

Nm/ r 2 + 1 ]

from

( 67b )

and

(67d):

A.-R.

200

Baghai-Wadji

(2),I Afj.r: 2

l T l

-( A

1+ fl

& E. Li

+/+ i

(2) 1

\r2)

(s)'

1 B,(2) 771 I T + T

Qi

(69a)

Eon I + y*im(0i,4i)

4(2)^

1

l+i

(S)

+ -*-

5,(2) 1

-(A

4(2) , lmr2

A

1+1

o(2)

92

1

(69b)

£0^2 J + 1

Combining the terms at the left-hand side of these equations, we can write (69) in the following compact form:

1 +

A{2)rl

(+1

1

^ITT^m^l^l) (70)

D(2)

•1 + i + i

1

Y

.^m rm(^h)

Multiplying both sides of (70) by (/ -I-1)/(2/ + 1) we obtain:

0

1

1

0

A

lmr2

(2) 1 B,lm 7+i"

sSra+r^^i^i)' e?r22! +

(71)

y

i i m (02,2)

Comment: It is interesting to note that the equation for the determination of the A—coefficient decouples from the the equation for the determination of the B—coefficient. As will be shown in the next section, this decoupling takes place irrespective of the number of layers. Since the matrix at the LHS is normal we obtain:

Factorization

Im

£•

of Potential and Field

0

1

1

0

201

Distributions

sTn2l+lYim(0^M (72)

B

l

(2) 1

Finally writing (67a) and (67c) in matrix form we have

lmT\

B,

4(2V

(ft)'

A

A

lmr2

(73)

(2) 1

(3) 1

(ft)'

Bl

l

->

The following comment completes the calculation of the unknown coefficients in our problem. Comment: The matrix at the RHS of (73) has the eigenvalues

A* = | H ] ± 1.

(74)

The corresponding normalized eigenvectors are s/2 2

±

(75)

X^ ^

2

Therefore, the matrix at the RHS of (73) can be decomposed as follows:

V2 2 V2 2

\/2 2 2

(ft)' + 1 0

V2 2

V2 2

s/2

_\f2

(76)

(ft)'-

Since the eigenvector matrix in (76) is normal, substituting the decomposition (76) for the matrix in (73) and multiplying the resulting equation by the inverse of the eigenvector matrix we obtain:

A.-R.

202

Baghai-Wadji

& E. Li

f ( 4 ^ + *, 3*0 (77)

0 5.2.

fe)'"

{2) l (2) Jl2 (A r - B ,1 ^ ^!mr2 ^m^+Ty

Explicit Formulae for the Potential Regions I, II, and III

Functions

in

In order to construct formulae for the potential functions in regions r < r±, r\
Substituting A ^ r j coefncients:

j+i

\e0ri2I

(78b)

and Bi>„/ r i + 1 into (73) we obtain the remaining

(79a

+

(S)'fe27TT^<"''«}

<79b>

Substituting these coefficients into (34a), (35a), and (36a) we obtain the formulae for ip^(r,9,<j>), tp^(r,6,4>), and ), respectively.

Factorization of Potential and Field Distributions

q2 r

$> 2

1 21+1

203

A (2) r

WV

A

lm 2

A

lmrl

l

r

(V^) 1 Vi

1 21+1

^

XniW

Fig. 7. Network representation of the algorithm for the calculation of the expansion coefficients ^ j ^ n in the 1st layer, and M4;„ir2 m * n e 2nd layer.

qi

V,

1 21+1

! =. 1n3(2) lm 1+1 r

>4W (ri/r 2 )

*2

V2

1 21+1

s> v.J

WV

3) - "nl( m

1

1+1 2

r

Fig. 8. Network representation of the algorithm for the calculation of the expansion coefficients •Bj m /»i M in the 2nd layer, and •Bj m /r 2 + in the 3rd layer. However, the major point here is something else: We want to investigate the forms of these expressions for the case r\ = r 2 . This corresponds to the problem of determining the potentials due to two point charges q\ and qi located at the points (n,9i,2) on the same n—sphere. W i t h n = r 2 (78) and (79) give: (2) _

92

B

%

Substituting A [ ^ r 2

=

(80a)

WTiVj1^2''

AY' = eon ^

^

+

and s } m / r i + 1

l

Y

^

^

(80b)

into (73) we obtain the remaining

204

A.-R.

Baghai-Wadji

& E. Li

coefficients:

.(1) _

92

1

1 v*

gi

1

1

Im

(3) _

B% =

g2

2fTT^e-M

(81a)

1

(81b)

Substituting for U ^ l and [ s ^ l from (78) into (35a), we obtain

OO

I

r'y, m (0,0) L

Z=0 m=—l

+

(82)

l

gi

1

j+ + 1i v «

r ri

eon 2Z + 1

y^((9i,0i)

^m(M)}

Recall that region II is defined by rx < r < r2. On the other hand we assumed that ri = r2- Therefore, in the present case, region II consists of the sphere r = r\. Considering this fact in (82) we obtain:

OO

(2)


I

M . « = 1=0E Em=-l{ +

J2

1_J_

eon 2Z + 1 r[

ycmah,) (83)

^2iT-^+lY^^

(Note the appearance of n at the LHS.) Next substituting (79a) and (79b) into (34a) and (36a), respectively, we have

Factorization of Potential and Field Distributions «2

1

1

,

v ^(r,e,t) = E E [ i^jaTT^to.fc)

(84a)

l

(=0 m=—l

' OO

205

1 ^ -r\Yl*rn(Oi,(t>i)}rlYlm(d,
£

^>M^) = E E_[ ^ a r r ^ 1 ^ ^ ^ )

< 84b )

1=0 m=-l

+

9i

A comparison with potential functions in (61) shows that we have been obtained the same expressions. But what about the formula in (83)? Setting r = ri in (84) we obtain OO

I

-.

-

= E E [ ikwrr^6^

VVM*)

^ = 0 771=—/

+ OO

(85a)

*

9i 1 1 e0n 21 + 1 ri

^WTiAY^Mr^^

/

r 1+1 ( <»>(rlfM) = E E .[ eiV27TT ' ^ ^^2) 0 ri 2/ + 1

(85b)

+ 91

1 +1 I ri F z ^(e 1) 0 1 )] ; i T y Im (6l^) e 0 ri 2/ +

and thus ^

( n , 0,0) = ^ ( 2 ) (r1,0,) = ¥>{3) (ri, M ) •

(86)

This means that <^2) (ri, 6, ) is consistent with and equal to the limits of ) for r —> ri. Therefore, since the potential function ) w e c a n omit it without any loss of information. 6. Five-Layer Problem In order to capture all features which are necessary to obtain recurrence formulae for the construction of potential functions we need to consider the

206

A.-R.

Baghai-Wadji

& E. Li

problem of calculating the potential functions associated with four point charges which lead to a five-layer problem. In what follows we rely on our results from previous sections but provide sufficient information to have a proper understanding of the construction procedure.

6.1. Point

Charges

Assume four point charges specified as follows:

Q{iHr,e,4>) =q~5(r-ri)6(cos0-cosei)6(-i)

(87)

i = 1,2,3,4

The radial distances of the point charges from the center of a chosen spherical co-ordinate partition the entire space into five regions. The electric potential and the radial component of the dielectric displacement in ith region are denoted by the superscript (i). We use the following correspondences:

Region I: r < rx $(D ^ A\yi DW

(88a)

<=> -eQUy

(88b)

*W**^+B%±

(89a)

Region II: r\ < r < r2

^^-^V+so^l^^r

Region III: r^ < r < r%

(89b)

Factorization

$(3)

of Potential and Field

Distributions

1 A

lmr

+ i 3

iim

(90a)

rl+l

(3) J . -60-^™ ^ +eo

207

J + l (3) 1 S m l +l

(90b)

r

Region IV: r$ < r < r±

(91a) (91b)

Region V: r$ < r

$(5) .

(5) B,lm

£o

6.2. Interface

1

(92a)

rl+l

(92b)

B ('m r i + l

Conditions

Satisfying the interface conditions on the r,—spheres (i obtain the following set of eight equations.

1,2,3,4) we

Interface Conditions on the n— Sphere:

KM]=A«r<1 +

B,(2)J_

(93a)

*m J+l '1

(2) [ ^ r i ] - M « r i + (i + l) B;im gl y«m(«l,0l) £0^1

Interface Conditions on the T2~Sphere:

i

J+l r l

(93b)

A.-R.

208

[^im^J +

D

lm

Baghai-Wadji

l+l —

& E. Li

A

lmr2

+

l[^2ri}-(l+l)B^-lA: + (J + 1)

D(3)_J_ •'lm l+l

(3) B,lm

i

l+l '2

(94a)

0)_J

lm'2

Q2

(94b)

£0^2

Interface Conditions on the 7-3—Sphere:

["Wl

L

1 o(3) *

J

-AWrl

+

}(4)_L_

(95a)

To

3

(4)_J_

+(J + i) 5,im

(+1

93 £0^3

^(^3,03)

(95b)

Interface Conditions on the r4—Sphere:

U ( 4 V1 4 - R ( 4 ) - l _ - B(5)

—

(96a)

lm r 1+1 4

4

94

e 0 r 4 y«m(«4,^4)

(96b)

We multiply the A— and 5—coefficients in various layers by appropriate non-dimensional quantities in order to obtain [J4)^T^] (S = 1,2,3,4) and [-^Jm/rs-i] ( s = 2,3,4,5) as the new coefficients. Furthermore, in each group of the interface conditions, we divide the second equation by I + 1. We obtain:

Factorization

of Potential

and Field

Distributions

AWrl] - feY Ll (2 Vl + \B{2) — A r lm \\ ~ y^J [Almr2\ + [^JmrJ+l T+i H<mVlJ ~ T+i W r*™1*2! j i Yt L B 1 + e0ri I + :

2&\-£m '<*M M-i

[^] ? (3)

+ 3(m

+

r

(2)

s;Im

(s

J+l

-® ^

(2) B;Im

J+l

X

1 J+l

1+1

[ B, "VJ = KM] + W + Bj'Im

£0»"2 i + 1

i

^2

Z + l Vr3

^(fc.fc)

i+i

[4M]

~2

1 J+l

i+i

r

? (3)

i

'Im J + l

,(4)J_1 7m i + i J

*Im(*3,&) e0r3 Z + 1

»*3

Z + l V>"4

A.-R.

210

KM] + (*)'

Baghai-Wadji

B (4) m J r

3

& E. Li

(5)J_ B,lm l+l

+1

(100a)

r

4

B,( 4 ) _ | _ r

3

+

1

(5) S im J+1

(100b)

£0^4 I + 1

Define

1 e0rs I + 1 *£,(*..*.) s = 1,2,3,4.

6(s)

(101)

Written in matrix form we have

A

lmr2

KM] = [(S)'

(102a) B,,(2) 7m

J^

4 (2) J

7Tl)[^]+ -*fe)' *

5, (2)_L_ lm

(i)

Q[

(102b)

211

Factorization of Potential and Field Distributions 3 4l(m V r 3

AAW-1 r

A

lm 2

1+1

(*)

B,(2)

1

(103a)

(*)'

,

(3)

1

5,Zm r*+

A
\r2)

R (2)

1

A(3)rl ^Im/'i

+

i +fii

<2, (2)

Vr3y

(103b)

? (3)

1 'lmZP?Z

4(3V

A

lmri

^imr3

1+1

te)

(104a)

- (s)

,(3)J_ Zm r i

(4) B,Zm

1 l r+ l

4(3) r l

'

/'r V

+1 ? (3)

~ \rij

i+T

1

*i™;p_, A

lmr4

1+1 \r4J

B

(104b)

Q, (3)

(4) 1

4(4) r I i+i

(S)

o(4)

,1+1

1+1

\r4J

B,1(5) ™J

1

4(4) r l o(4)

1

i +l

(5)

+ Bl

Substituting (102a) into (102b) we obtain

(105a)

L

= Q\42

(105b)

A.-R. Baghai-Wadji & E. Li

212

4 (2 V [0

1]

A

lmr2

R (2)

]±LQW

(106)

1

Substituting (105a) into (105b) we obtain

A\

;i

o]

(4U

(4) 1

B,

I + 1 Q(4) 2/ + l V / m

(107)

The remaining four equations in (103) and (104) written in matrix form give:

A{2)rl

(s) m -(s)'

(2) 1 1 B lm ri^"

(108) 0)^3

A,1V

(3) 1 1 B lm -J+

4 (2 V A

imr2

o(2) _ 1

(^J

-6(2)

'+i l^j ^1771^3

B (3) 1

(109)

Factorization

of Potential and Field

213

Distributions

A{3)rl A

lmr3 1

D(3)

1

(-)'+1

-(-)'

-1 A

= 0

(110)

= 4(3)

(111)

r

lm i

o(4) 1 lm^P^

D

r

3

J

4™M (3) 1

I l+l

'r y

+i

B,Zm

(4

,4 V 5,

Combining our results we obtain

T^+i

(4) 1

™lm

214

A.-R.

Baghai-Wadji

& E. Li

0 -1

(S)

"(S)

J+l

J+l 1+1 Vr3/

1 \M-i

( T3 J (+1

Os)

i+i

—j-M' ;+i V^y

o lm 2.

JiLoW 2i+l^m

1 B i(2) mT+T A4

(3)-I imr3

Q, (2)

(112) B (3) r

1 2

(4U AIV (m'4 (4) 1

B,

6(3)

J+inW

3

6.3. Decoupling

of the Coefficient

Matrix

In what follows we will demonstrate that the above system of equations decouples into two systems. Consider the first three equations:

Factorization

of Potential and Field

1

0

Distributions

01

0

J±Lod) 1

D(2)

/

\l+l

fe) -fe)

1

/

I l+l

xi+1

-fe)'

215

2l+l^
r

l

-1

0 Im o

-J-(a.)1

(2)

1

M-l ^ r 3 / /

D(3)

1

L Q«.

->lm 7 + T

(113) Adding the second equation to the third one, and also, multiplying the second equation by —1/(1 + 1) and adding the resulting equation to the third one we obtain:

1 + -*x

^

(' + * ) ( * ) '

i+i

0

- ( ! + £ ) (r

2

\i+i J

» 1

^ M-l

A%r< R (2)

1

A ^ 4(2)

^(mr3

B,

(3) 1

1 (114)

^lm

4(2)

Multiplying the second and third equations by (2Z + 1)/(Z +1) we obtain:

A.-R.

216

Baghai-Wadji

& E. Li

A

lmr2

0"

0(1)

D(2) 1 13 Im ZJTT

l+l 21 + 1

0

-(s)'

A(3)rl

(+1

Qlm

(115)

(2)

(S)

1

o(3) L

Ql

1 r

2

J

Finally multiplying the first equation by (ri/r2)l+1 sulting equation to the third equation we obtain:

and adding the re-

4 (2 V A

0

1

0

lmT2

Q, (1)

o(2) 1 B ImTFT

1 0

0 0

-(g) 0

•Almr3

,0)

l+l 21+1

Q?] fc

1

(-)

1+1 -

m+Q% (116)

Obviously the first and the third equations, which involve [•Bjm'/ri+1] and [•Bjm/r2+1]i decouple from the second equation, which interrelates

[A\y2] and {A\3M}. Next we consider the third equation in (116) together with the fourth and the fifth equations in (112):

Factorization

of Potential and Field

Distributions

217

Alm{3)r3rl

fi

1 7+T"

D(3)

n

lm

-1

(S)' I l+l

4(4V

-00'

B,(4)

1 3

V 2/ ft [(5

2Z+1

r

i1m m

^

^ ^lm

(117)

Qlm

Adding the second equation to the third one, and subtracting 1/(1 + 1) times the second equation from the third equation we obtain:

A?2r{ (3) 1 B,ImTTTT

21+1 (rA1 l+l \r4)

21+1 i+i

Alm(A)rirl

A

1

_2l±l

(rA

l+l

\r3)

2/4-1 l+l

B (4)

1

Q% +n(2)Ql

Ufe)

21+1

(3)

Q)

(118)

(1)

Q)

Multiplying the second and the third equations by (I + 1)/(2Z + 1) we obtain:

A.-R. Baghai-Wadji & E. Li

218

" 4 (3 V. " 0

1

A

lmr£

0'

0

o(3)

1

-(s)

0

1 r

2

0

A(4)rl

A

0

"(3)'

lmr4,

1

0

R(4)

1 r

L

3

J

fe)'+1^+C Z+ l 2/ + 1

(119)

3) g (Zm.

Q,(3) Multiplying the first equation by {r2/r^)l+1 t h e third equation we obtain:

and adding the result to

A

lmr3

0

0"

1

o(3)

1 '2

1

0

-05)'

0 A{A)rl

A

0

0

lmri

1_

r

L

,1+1 _

(£) f+ 1 21 + 1

3

(i)

J

(2)


0(3)

(120)

fe)mf(s)'+'«£+<5

n(2) lm

+ Q,(3)

Note t h a t replacing r a / r 4 by r 2 / r 3 we obtain the matrix in (116). Equations (116) and (120) taken together give:

Factorization of Potential and Field Distributions

0

1

0

0

0

0 R(2)

1

0

0

0

-(*)'

219

0

0

0

1

0

0

1

Im o

0

(3)

0

0

0

0

1 0

0 0

B,Im

2

-fe/ o 0

1 r^1

(4

4 V

1

(4)

B,

1

Q,(1)

J+ l 2/ + 1

\r2j

Ql™. + Qlm Q,(3)

'(%)WQ%

(3)

+ Q% + Q\

So far we have not used the last equation in (112). Adding this equation t o (121) results in:

220

A.-R.

0

1

1

0

0

0

0

0

Baghai-Wadji

& E. Li

0

0

0

0

0

0

0

1

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

1

0

-(*)'

-(*)'

0

(2U 1 A)t>r'. lm'2 (2) B lm

1

+1

r^

A*. (3) B lm

1 r1"^1

AfiV: (4) l B lm r

1 +1

(i)

Q)

(2)

l+ l 21 + 1

(-)

Qtt + Q% (122) Q,

(3)

^ lm

terferc+os

(3)

+Q

o{4)

Evidently this system of equations decouples into a subsystem for the determination of the coefficients A and a subsystem for the calculation of the coefficients B. The system for the B coefficients is immediately solvable:

Factorization

of Potential and Field 1

"D(2)

1

0

0"

0

1

0

0

0

1.

n

lmZTT

R(3)

221

•

l+l 21 + 1

1 r

2

o(4) 1 lmZTTT r 3

Distributions

n

L

J (i)

(2)

(S)

«+<5lm

fe)'+1[fe)'+1C

+ Ql

(123) (3)

+ QI

The system for the coefficients A is:

r>i (2) r'i

0

"(S)'

A

1

(*)'

1+1 6(3) 21 + 1

(124)

a(4)

AWri

1

0

lmrZ

Q{2)

Multiplying the third equation by {vs/r^)1 and adding the resulting equation to the second equation we obtain:

(9)'

o(2)

0

1

0

0

1_

lm

Alm{3)r3rl

/i

&

l+l l 21 + 1 &) Q.

AWri lAlmr4J

Multiplying the second equation by (r2/r3)1 equation to the first equation results in:

(4) , lm ^

Q(3)

**lm

(125)

0(4) **lm

and adding the resulting

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& E. Li

+c

1

0

o-

A

0

1

0

A{3)rl

0

0

1.

AWrl

lmr2

l+l 21 + 1

(3)

Qlm

(126)

For the determination of the remaining coefficients, i.e., [^(J^i] and [B^/r{+1} we use (102a) and (105a):

(127a) 1

?(5) r

(4)_J_

(127b)

4

Substituting for the coefficients A and B at the RHSs of (127) from (126) and (123) we obtain:

1+ ^2 r

^Im

3,

^Im

+^K^

(128a)

i+l

2/ + 1 (5) 1 B\Im 1+1 Z+ l

-:s)

+

Hi

HI

(S)

Qlm + Qlm

S(3)

(4)

+ <0+<&™

Summarizing our results we obtain the following formulae:

(128b)

Factorization

2l

+l

of Potential and Field

223

Distributions

\AWri

l+l H-ri.

+ <&>+$£!

~ J Qlm + Qlm

-\Ai2 1 + 1' ]A, H™ " r-

^Qtt + Q?

ITT n-^J " \u J $™ +

+ Q?2

r[^]=<5.n(4)lm

T+T

1

(2) B,(m

(129e)

\ l+l

21 + 1 (3)J_ B\ l+l

2Z + 1 i+ 1

(129d)

= Q,(1)

7+1

~ J

(129f)

Qlm + Qlm

1+1

B (4) _L r

(129b)

(129c)

^

21 + 1 1+

21 + 1

(129a)

3

l+l

(-

VoS+gf J Qlm "•" ^ i m +C

2/ + 1 (5) B:im /+ 1

(129g)

l+l

1

(+1

J* J \l + l

~ J

Qlm + Qlm + &

+ Qlm{4)

(129h)

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From the above we obtain the following recurrence formulae for the coefficients A and B:

(130a)

KM] = £) [ ^ ]

(2) 1 B 'm 1+1

i+i

2

(4)_j_

5 im

J+l

B,(4)_£_

(130e)

2/ + l ^ ' m

(3)_|_

B\'"i ~ ' + l r

(130d)

2Z + l v ' m

(2)_i_ l+l

B,lm

r2j

i+i

2l +

l^lm

(3) _ j _ ( 3 ) B\'"» J + l + - ^ 1 Qv m 2Z + l '

(130f)

(130g)

(130h)

Factorization

of Potential and Field

Distributions

225

- A(V

q4

V4

A

lmr4

(r3/r4)

V3

-e-

-

A(3) r1

A

lmr3

(r2/r3) q2

5>r2

-e-

A(2) r

J

*

A

r

-

A (1) r'

lm 2

(ri/r 2 )

r

*o l

- #

^

A

lmri

Fig. 9. Network representation of the algorithm for the iterative calculation of the expansion coefficients Avlri in i layer, i = N — 1, • • • , 1 .

7. Conclusion We presented a recipe for the factorization of potential- and field distributions without utilizing the Addition Theorem. Factorization plays a fundamental role in the majority of standard fast algorithms, e.g. the Fast Multipole Method and its variations. Factorization, as used in this paper, refers to the multiplicative decomposition (tensor product) of the potential functions into functions, which separately depend on the co-ordinates of the source- and the observation points. The multiplicative decomposition is crucial whenever we need to, e.g. integrate over a large number of source points, while keeping the observation point invariant. Such circum-

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(2) 1

qi

•*-

B lm

r

1+1 1

1+1

(r,/r2) q2

V2

(3) R _L 15

m

lm

1+1

1+1

(^3)

q3 ^ 3

^ 4

Blm

r

1+1

3

(r 3 /r 4 ) q4

(4) 1

e1+1

(5) 1

•e-

Blm

r

l+l

4

Fig. 10. Network representation of the algorithm for the iterative calculation of the expansion coefficients -B;„/»"'iJ hi i t h layer, i = 2, • • • ,N.

stances routinely arise in large scale engineering computations based on singular surface integrals. T h e proposed factorization relies on the diagonalization of the underlying partial differential equations with respect to a distinguished direction in space. We outlined our method in great detail by introducing a constructive recipe for the diagonalization of the Poisson's equation in spherical co-ordinates. We provided a self-sufficient exposition on the factorization of potential- and field functions. It can be shown that the proposed recipe is applicable to problems in electrodynamics and elastodynamics, involving isotropic, anisotropic, or bianisotropic materials. More generally, our recipe is applicable to any system of linear partial differential equations which can be diagonalized. Recently

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we published a conjectured claiming that all linear physically realizable systems belong to this class of problems. Therefore, it is reasonable to expect t h a t the proposed method will enable us to apply the Fast Multipole Method to those problem in engineering practice for which no Addition Theorem has been formulated so far. To enhance the efficiency of computations further, wavelets [19]-[22], and problem-specific wavelets [23]-[25] can be used to obtain sparse "impedance matrices" in the Boundary Element Method applications. Wavelets and frames based on radial functions seem to be very promising tools in solving boundary value problems. On the other hand, the functional dependence of the factorized form in cylindrical- and spherical co-ordinates involves radial functions. We are presently investigating the construction of wavelets and frames which are induced by the radial functions which appear in multipole expansions. Acknowledgement This work was done while one of the authors (ARBW) was visiting the Institute for Mathematical Sciences, National University of Singapore and Institute of High Performance Computing (IHPC) in 2003. The visit was supported by the Institute and IHPC. References 1. W. D. Jackson, "Classical Electrodynamic," John Wiley & Sons, Inc., New York, 1975. 2. L. Greengard, "The Rapid Evaluation of Potential Fields in Particle Systems," MIT, Cambridge, 1987. 3. V. Rokhlin, "Rapid Solution of Integral Equations of Scattering Theory in Two Dimensions," J. Comput. Phys., vol. 86, no. 2, pp. 414-439, 1990. 4. R. Coifman, V. Rokhlin, and S. Wandzura, "The Fast Multipole Method for the Wave Equation: A Pedestrian Prescription," IEEE Antennas Propagat. Mag., vol. 35, no. 3, pp. 7-12, June 1993. 5. C. C. Lu and W. C. Chew, "A Fast Algorithm for Solving Hybrid Integral Equation," IEE Proceedings-H, vol. 140, no. 6, pp. 455-460, December 1993. 6. N.I. Muskhelishvili, "Singular Integral Equations," P. Noordhoff N.V. Groningen - Holland, 1953. 7. C. A. Brebbia, J. C. F. Telles, and L. C. Wroble, "Boundary Element Techniques," Springer Verlag, 1984. 8. R. F. Harrington, "Field Computation by Moment Methods," New York: Macmillan, 1968. 9. M. M. Ney, "Method of Moments as Applied to Electromagnetic Problems," IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 972-980, 1985.

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10. R. P. Kanwal, "Generalized Functions: Theory and Technique," Academic Press, 1983. 11. A. R. Baghai-Wadji, " A Unified Approach for Construction of Green's Funct i o n s , " Habilitation Thesis, Vienna University of Technology, 1994. 12. A. R. Baghai-Wadji, " T h e o r y and Applications of Green's Functions," a chapter in Selected Topics in Electronics and Systems - Vol. 20: Advances in Surface Acoustic Wave Technology, Systems and Applications (Vol. 2), C. C.W. Ruppel, and T. A. Fjeldly (Editors), p p . 83-149, World Scientific, 2001. 13. A. R. Baghai-Wadji, "Fast-MoM: A Method-of-Moments Formulation for Fast C o m p u t a t i o n s , " in A C E S Journal, special issue of t h e Applied Computational Electromagnetics Society Journal, pp. 75-80, vol. 12, no. 2, 1997. 14. A. R. Baghai-Wadji a n d D. Penunuri, "Coordinate-Free, FrequencyIndependent Universal Functions for BAW Analysis in SAW Devices," in Proc. I E E E Ultrosonics Symp., pp. 287-290, 1995. 15. S. Ramberger, and A.R. Baghai-Wadji, " E x a c t Eigenvector- and Eigenvalue Derivatives with Applications to Asymptotic Green's Functions," Proc. A C E S 2002, T h e 1 8 t h Annual Review of P r ogress in Applied C o m p u t a t i o n a l Electromagnetics, March 18-22. Monterey, USA. 16. K. Varis, and A.R. Baghai-Wadji, "Pseudo-Spectral Analysis of Radially-Diagonalized Maxwell's Equations in Cylindrical Co-ordinates," Optics Express, vol. 11, no. 23, p p . 3048-3062, 2003. 17. M. T. Manzuri-Shalmani, and A.R. Baghai-Wadji, "Elemental Field Distributions in Corrugated Structures with Large-Amplitude G r a t i n g s , " Electronics Letters, 2003, accepted. 18. A. R. Baghai-Wadji, a book chapter, " A Symbolic Procedure for t h e Diagonalization of Linear P D E s in Accelerated Computational Engineering," 15 pages, Editors: Franz Winkler and Ulrich Langer, Springer Verlag, 2003. 19. Y. Meyer, "Ondelettes et Operateurs, I: Ondelettes," Hermann, 1990. English translation: "Wavelets and O p e r a t o r s , " Cambridge University Press, 1992. 20. I. Daubechies, " T e n Lectures on Wavelets," CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1992. 21. G.G. Walter, "Wavelets and Other Orthogonal Systems with Applications," C R C Press, Inc., 1994. 22. E. Hernandez, anf G. Weiss, "A First Course on wavelets," C R C Press, Inc., 1996. 23. A.R. Baghai-Wadji, and G. G. Walter, "Green's Function-Based Wavelets," International IEEE-SU, Sonics and Ultrasonics Conference, P u e r t o Rico, 2000. 24. A.R. Baghai-Wadji, and G. G. Walter, "Green's Function-Based Wavelets: Selected P r o p e r t i e s , " International IEEE-SU, Sonics and Ultrasonics Conference, P u e r t o Rico, 2000. 25. A.R. Baghai-Wadji, and G. G. Walter, "Green's Function Induced Waveletsand Wavelet-like Orthogonal Systems for EM Applications," Proc. A C E S 2002, T h e 1 8 t h Annual Review of Progress in Applied Computational Electromagnetics, March 18-22. Monterey, USA.

VIRTUALIZATION-AWARE APPLICATION FRAMEWORK FOR HIERARCHICAL MULTISCALE SIMULATIONS ON A GRID Aiichiro Nakano, Rajiv K. Kalia, Ashish Sharma, Priya Vashishta Collaborator)) for Advanced Computing and Simulations, Department of Computer Science, Department of Physics & Astronomy, Department of Materials Science & Engineering, University of Southern California Los Angeles, CA 90089-0242, USA E-mail: (anakano, rkalia, sharmaa, priyav)@usc.edu Shuji Ogata Graduate School of Engineering, Nagoya Institute of Technology Nagoya 466-8555, Japan E-mail: [email protected] Fuyuki Shimojo Department of Physics, Kumamoto University Kumamoto 860, Japan E-mail: [email protected] A virtualization-aware application framework is developed, based on datalocality principles, to perform hierarchical multiscale simulations of materials on a Grid of distributed computing and visualization platforms. The framework combines: linear-scaling algorithms based on space-time multiresolution techniques; topology-preserving computational-space decomposition with wavelet-based adaptive load balancing; and immersive and interactive visualization of billion-atom datasets. Multiscale simulations are performed on a Grid to seamlessly integrate quantum mechanical calculation based on the density functional theory and atomistic simulation based on the molecular dynamics method.

1. Introduction Metacomputing on a Grid1 of geographically distributed Teraflop-toPetaflop computers and immersive virtual reality environments connected via high-speed networks will revolutionize science and engineering, by enabling hybrid simulations that integrate multiple expertise distributed globally.2 Such multidisciplinary applications are

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emerging at the forefront of computational science and engineering. For example, hierarchical multiscale simulations embed accurate calculations—e.g., quantum mechanical (QM) calculations to handle chemical reactions—within coarse models—e.g., molecular dynamics (MD) simulation to describe large-scale atomistic processes and finite element (FE) calculation for continuum mechanics—only where/when high-fidelity modeling is required.3"5 On a Grid, applications will likely be virtualized, i.e., the user will not know on which computers the code is running. However, virtualization of high-end applications (such as billion-atom MD and thousand-atom QM simulations) is an unsolved challenge. To support virtualization, applications need to be: 1) scalable from a single processor to thousands of processors; 2) portable from one architecture to another in terms of performance; and 3) adaptive to dynamically changing computing resources. In the past years, we have been developing a virtualization aware application framework for hierarchical multiscale materials simulations, based on data locality principles. Our framework includes 0(N) algorithms (N is the problem size) for broad applications such as: 1) space-time multiresolution molecular dynamics (MRMD) algorithm for large-scale atomistic simulations; and 2) linear-scaling density functional theory (LSDFT) algorithm to reduce the exponential complexity of the QM problem. To implement these algorithms on massively parallel and distributed computing platforms, our framework also includes: 1) topology-preserving computationalspace decomposition for structured message passing to minimize the latency; and 2) wavelet-based curvilinear-coordinate load balancing to minimize the load-imbalance and communication costs. Finally, the framework enables interactive visualization of high-dimensional datasets from extreme-scale (e.g., billion atoms) simulations in immersive virtual environments, using multilevel, probabilistic and parallel/distributed techniques. In Sec. 2, we describe the 0(N) MRMD and LSDFT algorithms, and Sec. 3 deals with the scalable parallel computing framework. Multiscale simulations and immersive/interactive visualization techniques are presented in Sees. 4 and 5, respectively. Finally, Sec. 6 contains conclusions.

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2. Linear-Scaling Atomistic Simulation Algorithms Spatiotemporal localities inherent in scientific/engineering problems can be utilized to design algorithms with low computational complexities. This section shows two such examples in MD and QM simulations. lA.Multiresolution Molecular Dynamics In the MD approach, one obtains the phase-space trajectories of the system (positions and velocities of all atoms at all time) by numerically integrating Newton's equations of motion,6'7 dr

dr{

\ I

where M, and r, are the mass and position of the z'-th atom, respectively. In Eq. (1), atomic force laws for describing how atoms interact with each other is encoded in the interatomic potential energy, EMo(rN), which is a function of the positions of all N atoms, r^ = {ri, r2,..., r^}. In our manybody interatomic potential scheme,8 EMDO^) is expressed as an analytic function that depends on relative positions of atomic pairs and triplets. The hardest computation in MD simulations is the evaluation of long-range electrostatic forces, which requires 0(N2) operations. The multiresolution molecular dynamics (MRMD) algorithm6'7'9 reduces this 0(N2) complexity to 0(N), by making use of spatial localities based on the fast multipole method (FMM) 10 ' u . The first essential idea of the FMM is clustering, i.e., instead of computing interactions between all atomic pairs, atoms are clustered and cluster-cluster interactions are computed (see Fig. 1). At the source of interaction, cluster information is encapsulated in terms of the multipoles of the charge distribution with a well-defined error bound. At the destination, the electrostatic potential is expanded in terms of local terms, which is similar to the Taylor expansion. The second essential idea is to use larger clusters for longer distances, in order to reduce the computation and keep the error constant. This is achieved by recursively subdividing the simulation box into smaller cells to form an octree data structure. The O(N) algorithm traverses this tree twice. In the upward pass, multipoles are computed for all cells at all levels. First the multipoles of the leaf cells are computed using atomic charges and coordinates, and the multipoles of these children cells are shifted and combined to obtain the multipoles of the parent cells. This procedure is

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repeated until the root of the octree is reached. In the downward pass, these multipoles are translated to local terms for all cells at all levels, starting from the root. For a given cell at each level, only the multipoles of a constant number of interactive cells contribute to the local terms. Contributions from farther cells have already been computed at the previous coarse level, and they are inherited from the parent cell. On the other hand, the contributions from the nearest-neighbor cells will be computed at the next fine level. This procedure is repeated until we reach the leaf level. Finally, the nearest-neighbor-cell contributions at the leaf level are computed by direct summation over atoms. Since constant computation is performed at each of the O(N) octree nodes, the complexity of the FMM algorithm is O(N). Our scalable and portable parallel FMM algorithm, which has the unique capability to compute stress tensors with a novel complex charge method, is freely available from an open source public library.11

Fig. 1. Schematic of the FMM. (Left) Atoms (dots) are clustered (squares), and clustercluster interactions are computed. (Center) Two-dimensional example of hierarchical clustering, in which the entire system at level / = 0 is recursively divided into subsystems. (Right) For a given cell at an octree level (shown in black in the bottom panel), only the multipoles from a constant number of neighbor cells (shown in gray in the bottom panel) are translated to the local terms. Contributions from farther cells (hatched) have already been computed at the previous coarse level, and they are inherited from the parent cell (black in the top panel). On the other hand, translation of the multipoles from the nearestneighbor cells (white in the bottom panel) will be delegated to the children cells at the next fine level.

The MRMD algorithm also uses multiple time stepping (MTS)9'12' 13 to take advantage of temporal localities. The MTS uses different forceupdate schedules for different force components, i.e., forces from the nearest-neighbor atoms are computed at every MD step, and forces from

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farther atoms are computed with less frequency. This not only reduces the computational cost but also enhances the data locality, and accordingly the parallel efficiency is increased. These different force components are combined using a reversible symplectic integrator,12 and the resulting algorithm consists of nested loops to use forces from different spatial regions. It has been proven that the phase-space volume occupied by atoms is a simulation-loop invariant in this algorithm, and this loop invariant results in excellent long-time stability of the solutions. 2.2. Linear-Scaling Density Functional Theory on Hierarchical Grids Breakage and formation of bonds in chemically reactive regions need to be described by a QM method, which explicitly treats the electronic degrees-of-freedom. Since each electron's wave function is a linear combination of many states, the combinatorial solution space for the many-electron problem is exponentially large. The density functional theory (DFT) avoids the exhaustive enumeration of many-electron correlations by solving M single-electron problems in a common average environment (M is the number of independent wave functions and is on the order of JV).'4"16 As a result, the problem is reduced to a selfconsistent matrix eigenvalue problem, which can be solved with O(A^) operations. The DFT problem can also be formulated as a minimization of the energy, EQM(TN, I|>M), with respect to electron wave functions, tyM(r) = {xp ](r), ii>2(r), ..., i p ^ r ) } , subject to orthonormalization constraints between the wave functions. We include electron-ion interaction using norm-conserving pseudopotentials17' l8 and the exchange-correlation energy associated with electron-electron interaction in a generalized gradient approximation19. For efficient parallel implementation of DFT, we have developed a hierarchical real-space grid method,20' 21 based on higher-order finite differencing and multigrid acceleration ' ' . In the hierarchical grid method, these real-space multigrids are adaptively refined25 near each atom to accurately describe pseudopotentials (see Fig. 2). For larger systems (M > 1,000), however, the 0(M3) orthonormalization becomes the bottleneck, and hence, linear-scaling DFT algorithms become essential.26 We have implemented21 an O(M) algorithm27 based on the data locality principle called "quantum nearsightedness"28—the observation that, for most materials at most temperatures, the off-diagonal elements of the density matrix,

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p(r,r')=^^(r)^ / ! (r'),

(2)

n=l

decay exponentially, i.e., p(r,r')ocexp(-C l r - r ' l ) for lr-r'H»oo (Cis a constant). Such a diagonally dominant matrix can be represented by maximally localizing each wave function, ip„(r), by a unitary transformation and then truncating it with a finite cut-off radius. In addition, a Lagrange-multiplier-like technique is used to perform unconstrained minimization, avoiding the O(A^) orthonormalization procedure.

Fig. 2. Schematic of hierarchical real-space grids. Coarse multigrids (gray) are used to accelerate iterative solutions of the DFT problem. Fine grids (meshes in the bottom panel) are adaptively generated near the atoms (spheres in the bottom panel) to accurately operate pseudopotentials on the wave functions.

3. Scalable Parallel Computing Framework Data locality principles are key to developing a scalable parallel computing framework as well. For parallelization of MRMD and LSDFT algorithms, we have developed a topology-preserving computational spatial decomposition scheme6'7 to minimize latency through structured message passing9 and load-imbalance/communication costs through a novel wavelet-based load-balancing scheme29. In spatial decomposition, the total volume of the system is divided into P subsystems of equal volume, and each subsystem is assigned to a processor in an array of P processors. To calculate the force on an atom in a subsystem, the coordinates of the atoms in the boundaries of neighbor subsystems are "cached" from the corresponding processors. After updating the atomic positions due to a time-stepping procedure, some atoms may have moved

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out of its subsystem. These atoms are "migrated" to the proper neighbor processors. With the spatial decomposition, the computation scales as NIP while communication scales in proportion to (NIP)213 for an N-atom system. Many MD simulations are characterized by irregular atomic distribution and associated load imbalance. For irregular data structures, the number of atoms assigned to each processor varies significantly, and this load imbalance degrades the parallel efficiency. The load-balancing problem can be stated as an optimization problem, i.e., one minimizes the load-imbalance cost as well as the size and the number of messages: T = W.p(max p |{i I r, E p}\) + tcomm(maxpp +f

I ||r,- -dp\ < rc }|)

latency(maxp[^message(P)])

where the three terms are the load-imbalance cost, the size of messages, and the number of messages, respectively. In Eq. (3), dp and Nmessage{p) denote the boundary surface of the physical volume assigned to processor p, and the number of messages per MD step for/?, respectively. The expression, maxg/(p), denotes the maximum value of function f[p) over all the processors, and rc is the range of the interatomic potential. The prefactors, /comp, (Comm and latency, are constants related to the processor speed, communication bandwidth and latency, respectively, and they are determined experimentally by test runs on the parallel computer under consideration. To minimize the number of messages, we preserve the 3D mesh topology, so that message passing is performed in a structured way in only 6 steps. To minimize the load imbalance cost as well as the message size, we have developed a computational-space decomposition scheme.30 The main idea of this scheme is that the computational space shrinks where the workload density is high and expands where the density is low, so that the workload is uniformly distributed in the computational space. To implement the curved computational space, we introduce a curvilinear coordinate transformation, t = x + u(x),

(4)

where x is a position in the physical Euclidean space and u(x) is a deformation field. We then use regular 3D mesh topology in the computational space, \, to map atom / to processor p in an array of Px x Py x Pz processors:

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Pfa) = Px{Six)PyPz + Py (liy )PZ + P&iz) Pa{%ia) = [%iaPa/La\

'

... V. /

( « = *,V,z)

where §; = (|f«, ^ , ffe) is the coordinate of atom i and L a is the simulation box size in the a direction in the computational space. This regular 3D mesh partition in the computational space results in curved partition boundaries in the physical space, x. The load-imbalance and communication costs are minimized as a functional of the coordinate transformation, l(x), using simulated annealing. We have found that wavelet representation leads to compact representation of curved partition boundaries, and accordingly to fast convergence of the minimization procedure.29 Another critical issue in high-end parallel/distributed computing is the handling of large datasets. For example, a 1.5-billion-atom MD simulation we are currently performing produces 150 GB of data per time step (or every few seconds), including atomic species, positions, velocities, and stresses. For scalable input/output (I/O) of such large datasets, we have designed a data compression algorithm based on data localities.31 It uses octree indexing and sorts atoms accordingly on the resulting spacefilling curve, which is a mapping from the threedimensional space to a one-dimensional list, while preserving the spatial proximity of successive list elements. By storing differences between successive atomic coordinates, the I/O requirement for a given error tolerance reduces from 0(N\ogN) to 0(N). An adaptive, variable-length encoding scheme is used to make the scheme tolerant to outliers. An order-of-magnitude improvement in the I/O performance was achieved for actual MD data with user-controlled error bound. The 0(N) algorithms in Sec. 2 and the parallel computing framework in this section have been combined to achieve scalability up to 6.4 billion-atom MRMD and 0.44 million-electron LSDFT simulations on multi-Teraflop architectures.6 4. Hierarchical Multiscale Simulations on a Grid Our hierarchical multiscale simulation framework4' 5 consists of: i) hierarchical division of the physical system into subsystems of decreasing sizes and increasing quality-of-solution (QoSn) requirements (e.g., needs for describing nonlinear atomistic processes or chemical reactions), S0 D Si D ... D S„; and ii) a suite of simulation services, Ma (a

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= 0, 1, ..., n), of ascending order of accuracy (e.g., FE -< MD -< DFT), see Fig. 3. In this scheme, an accurate estimate of the energy of the entire system is obtained from the recurrence relation,32 E

a{Si)

= Ea-l{Si)

+ Ea{Si+\)-

E

a-\{Si+\)>

(6)

where EJJSi) is the energy of system S, calculated with method Ma. This modular, additive hybridization scheme has minimal interdependency and communication between simulation modules. Other physical quantities such as interatomic forces are obtained by derivatives of Eq. (6), and accordingly are additive as well. . Level of theory Accurate M 2 T O "

Coarse

Y Small

y^ System size Large

Fig. 3. Extrapolation in the two-dimensional meta-model space in a hierarchical multiscale simulation. Recursive applications of Eq. (6) accurately describe a large system (denoted by star), using less compute-intensive calculations (circles).

We have used the additive hybridization framework to perform: 1) MD/DFT simulations of crack initiation in Si in the presence of water molecules;33 2) FE/MD simulations of stress distributions at Si/amorphous Si3N4 interfaces;34 and 3) multiscale FE/MD/DFT simulations of oxidation in Si.4 We have also performed a multidisciplinary, collaborative MD/DFT simulation on a Grid of geographically distributed Linux clusters in the US and Japan, based on the modular, additive hybridization scheme (see Fig. 4).2 The multiscale MD/QM simulation code has been Grid-enabled based on a divide-and-conquer scheme, in which the QM region is a union of multiple QM clusters.

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Fig. 4. Multiscale MD/DFT simulation of the reaction of water at a crack tip in silicon (top), on a Grid of distributed Linux clusters in the US and Japan (bottom). In this figure, five QM calculations (circles) around five water molecules are embedded in an MD simulation.

Since the energy is a sum of the QM energy corrections for the clusters in the additive divide-and-conquer hybridization scheme, £ = £ M D (system) + X[ £ 'DFT( cluster )~ £ 'MD( cluster )]-

(7)

cluster

each QM-cluster calculation does not access the atomic coordinates in the other clusters, and accordingly its parallel implementation involves no inter-QM-cluster communication. Furthermore, the multiple-QMcluster scheme is computationally more efficient than the single-QMcluster scheme because of the O(A^) scaling. (The large prefactor of 0(N) DFT algorithms makes conventional 0(A^) algorithms faster below a few hundred atoms.21) The hybrid MD/DFT simulation algorithm has been implemented on parallel computers, by first dividing processors into the MD and DFT calculations (task decomposition) and then using spatial decomposition in each task. The additive hybridization scheme makes the MD and DFT subtasks entirely independent except for the exchange of cluster-atom coordinates and calculated forces. The MD processors compute the

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239

energy and forces of the entire system and send the atomic coordinates of the QM clusters to each of the QM processor groups. Subsequently, the MD and QM processors independently perform the MD and QM computations on the atomic clusters. The QM energy and forces are then returned to the MD processors, where the total energy and corresponding forces are calculated and the equations of motion are integrated to update the atomic positions and velocities. The communications between the MD and QM processors are minimal, since the MD processors only need to send several hundred atomic coordinates to each QM cluster, which in return sends back the calculated several hundred force components. We have implemented the multiscale MD/DFT simulation algorithm as a single MPI (Message Passing Interface)35 program. The Globus middleware1 and the Grid-enabled MPI implementation, MPICH-G2,36 have been used to implement the MPI-based multiscale MD/DFT simulation code in a Grid environment. In the initial implementation, processors on multiple PC clusters are statically allocated using a host file. The user specifies the number of processors for each QM-cluster calculation in a configuration file. In more recent MD/DFT simulations, a simple local error indicator based on atomic bond lengths has been used to automatically change the size of QM calculations in run-time. The Gridified MD/QM simulation code has been used to study environmental effects of water molecules on fracture in silicon. A preliminary run of the code has achieved a parallel efficiency of 94% on 25 PCs distributed over 3 PC clusters in the US and Japan, see Fig. 4. 5. Immersive and Interactive Visualization of Massive Datasets Data locality principles also play a critical role in designing scalable visualization techniques. Interactive visualization/mining of highdimensional datasets is essential for understanding hybrid multiscale material simulations.37 An immersive virtual environment is ideal for interactively exploring complex material processes, e.g., in nanoceramics and nanocomposites. We have an immersive and interactive visualization platform called ImmersaDesk, which is used to render billion-atom datasets at a near interactive speed. To achieve this capability, we have developed a visualization system based on a parallel and distributed approach with a Linux cluster to efficiently select a data subset within the field-of-view (view-frustum culling) using the octree data structure (see Fig. 5).38'39

240

A. Nakano et al.

Fig. 5. Schematic of octree-based view-frustum culling. (Left) Extraction of octree nodes (in gray) that contain the viewer's field-of-view (triangle). (Right) Recursive traversal of the octree to check if the nodes at different levels intersect with the field-of-view. If a parent node is found to be visible, then its children are tested. When applied to the entire octree, the visible nodes are extracted in CHlogN) time.

We have also developed a novel probabilistic approach to efficiently remove hidden atoms (occlusion culling) far from the viewer.38,39 This approach is based on a recurrence relation, vc-(l-Dc)vc_ls

(8)

where vc is the visibility (i.e., the fraction of atoms that are probably seen by the user) of the c-th octree cell and Dc is the density of atoms (normalized in the range between 0 and 1) of the c-th octree cell. In Eq. (8), the leaf octree cells are ordered in an ascending order of distance from the user, based on a line-drawing algorithm. When the viewer is moving, the probabilistic occlusion culling is activated to decimate atoms with probability 1 - vc, with typically a few percent of pixel loss. Furthermore, we use a machine-learning approach to predict the user's next movement and prefetch data from the Linux cluster to the graphics server.40 Using this visualization system, we have demonstrated interactive visualization of a billion-atom dataset in an immersive virtual reality environment. 6. Conclusion In summary, we are developing a virtualization aware framework for hierarchical multiscale simulations based on data locality principles.

Visualization-Aware

Application

Framework

241

Such a framework will become increasingly more important in the coming era of "cyber-infrastructures for science and engineering," when globally distributed multidisciplinary teams will collaborate on a Grid. The scope of hierarchical multiscale simulations is rapidly expanding. For example, we have recently performed multiscale FE/MD/DFT simulations to study the oxidation of silicon surface.4 Such hierarchical multiscale simulations will enhance the scalability on Grid architectures by performing more accurate but less scalable computations only when and where they are needed, and they are expected to play a significant role in scientific research and engineering developments in the cyber-infrastructure era. Acknowledgments This work was partially supported by AFRL, ARL, DOE, NSF, and USC-Berkeley-Princeton DURINT. Benchmark tests were performed at Department of Defense (DoD)'s Major Shared Resource Center under a DoD Challenge Project. Parallel simulations were also performed on the 1,512-processor HPC cluster at the Research Computing Facility and 400+ processor Linux clusters at the CoUaboratory for Advanced Computing and Simulations at the University of Southern California.

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References 2

3

4

5

6

7

8

9

10 11

12

13 14 15 16

17 18 19 20

21

22 23 24 25 26 27 28 29 30

I. Foster and C. Kesselman, The Grid 2: Blueprint for a New Computing Infrastructure (Morgan Kaufmann, San Francisco, 2003). H. Kikuchi, R. K. Kalia, A. Nakano, et al, Proc. Supercomputing '02 (IEEE, 2002). J. Q. Broughton, F. F. Abraham, N. Bernstein, et al., Phys. Rev. B 60, 2391 (1999). S. Ogata, E. Lidorikis, F. Shimojo, et al., Comput. Phys. Commun. 138, 143 (2001). A. Nakano, M. E. Bachlechner, R. K. Kalia, et al., Comput. Sci. Eng. 3(4), 56 (2001). A. Nakano, R. K. Kalia, P. Vashishta, et al., Scientific Programming 10, 263 (2002). A. Nakano, T. J. Campbell, R. K. Kalia, et al., in Handbook of Numerical Analysis, Special Volume on Computational Chemistry, ed. C. Le Bris (Elsevier, Amsterdam, 2003) p. 639. P. Vashishta, R. K. Kalia, and A. Nakano, J. Nanoparticle Research 5,119 (2003). A. Nakano, R. K. Kalia, and P. Vashishta, Comput. Phys. Commun. 83, 197 (1994). L. Greengard and V. Rokhlin, J. Comput. Phys. 73, 325 (1987). S. Ogata, T. J. Campbell, R. K. Kalia, et al., Comput. Phys. Commun. 153, 445 (2003). G. J. Martyna, M. E. Tuckerman, D. J. Tobias, et al, J. Chem. Phys. 101, 4177 (1994). A. Nakano, Comput. Phys. Commun. 105, 139 (1997). P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). W. Kohn and L. J. Sham, Phys. Rev. 140, Al 133 (1965). W. Kohn and P. Vashishta, in Inhomogeneous Electron Gas, eds. N. H. March and S. Lundqvist (Plenum, 1983) p. 79. N. Troullier and J. L. Martins, Phys. Rev. B 43, 8861 (1991). R. D. Kingsmith, M. C. Payne, and J. S. Lin, Phys. Rev. B 44, 13063 (1991). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). S. Ogata, F. Shimojo, R. K. Kalia, et al, Comput. Phys. Commun. 149, 30 (2002). F. Shimojo, R. K. Kalia, A. Nakano, et al, Comput. Phys. Commun. 140, 303 (2001). J. R. Chelikowsky, Y. Saad, S. Ogiit, et al., Phys. Stat. Sol. (b) 217, 173 (2000). J.-L. Fattebert and J. Bernholc, Phys. Rev. B 62, 1713 (2000). T. L. Beck, Rev. Mod. Phys. 72, 1041 (2000). T. Ono and K. Hirose, Phys. Rev. Lett. 82, 5016 (1999). S. Goedecker, Rev. Mod. Phys. 71, 1085 (1999). F. Mauri and G. Galli, Phys. Rev. B 50, 4316 (1994). W. Kohn, Phys. Rev. Lett. 76, 3168 (1996). A. Nakano, Concurrency: Practice and Experience 11, 343 (1999). A. Nakano and T. J. Campbell, Parallel Comput. 23, 1461 (1997).

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A. Omeltchenko, T. J. Campbell, R. K. Kalia, et al., Comput. Phys. Commun. 131, 78 (2000). S. Dapprich, I. Komaromi, K. S. Byun, et al., J. Mol. Struct. (Theochem) 461462, 1 (1999). S. Ogata, F. Shimojo, R. K. Kalia, et al., J. Appl. Phys. (2004) in press. E. Lidorikis, M. E. Bachlechner, R. K. Kalia, et al., Phys. Rev. Lett. 87, 086104 (2001). W. Gropp, E. Lusk, and A. Skjellum, Using MPI, 2nd Ed. (MIT Press, Cambridge, 1999). N. T. Karonis, B. Toonen, and I. Foster, J. Parallel Distributed Comput. 63, 551 (2003). A. Nakano and J. X. Chen, Comput. Sci. Eng. 5, 14 (2003). A. Sharma, A. Nakano, R. K. Kalia, et al., Presence-Teleoperators and Virtual Environments 12, 85 (2003). A. Sharma, R. K. Kalia, A. Nakano, et al, Comput. Sci. Eng. 5(2), 26 (2003). X. Liu, A. Sharma, P. Miller, et al., in Proc. Int'l Conf. on Parallel & Distributed Processing Techniques & Applications IV (CSREA Press, 2002) p. 2054.

M O L E C U L A R D Y N A M I C S SIMULATION A N D LOCAL QUANTITIES

Tamio Ikeshoji Research Institute for Computational Sciences (RICS), National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 2, Umezono 1-1-1, Tsukuba, 305-8568, Japan E-mail: [email protected] Various roles of molecular dynamics simulation are discussed, at first. Then, in order to bridge the macroscopic simulation and atomic level simulation, calculation methods of the local quantities, particularly local pressure, in the molecular dynamics simulation are discussed. It is necessary to integrate force between atoms along the path connecting the atoms within the local volume. This integration corresponds to the virial term used in the molecular dynamics calculation under the periodic boundary conditions. 1. Introduction 1.1. Equilibrium

and

non-equilibrium

Molecular dynamics (MD) gives trajectories of all atoms in the system simulated. MD is used and has been developed to calculate various physical properties relating to thermodynamics and kinetics under the equilibrium conditions and to calculate various processes under nonequilibrium conditions. 1.1.1. Thermodynamic

states

Under the equilibrium, time average is important in MD calculations. Therefore, time development is not necessary to be realistic. A lot of methods giving various thermodynamic states have been proposed. Nose's thermostat [1], Parrinello-Rahman's constant pressure algorithm [2], and so on are historically important and are generally used in the present MD simulations. Recently, some new ensemble methods like Multi-canonical ensemble

245

246

T. Ikeshoji

method have been presented and are being tested.

1.1.2. Kinetic parameters Some kinetic parameters can be obtained by equilibrium MD. Activation energy, for instance, is calculated by constrained MD, in which bond distance, configurations, etc. are constrained. Blue-moon sampling method [3] is an example using the constraints method. Efficient methods to simulate rare events have been recently presented; for instance, meta-dynamics method [5].

1.2. NEMD

(NonEquilibrium

Molecular

Dynamics)

Molecular dynamics has another aspect which concerns time development of systems. In this case, time development must be physically meaningful, though atomic trajectories may not be correct. MD simulations of fractures, phase transition processes, dislocations, etc. are in this category. This kind simulation can be called nonequilibrium MD (NEMD). But, it means usually simulations at the steady state under the non-equilibrium conditions; simulations of thermal flow, mass flow, and so on.

1.2.1. Boundary-driven

NEMD

From simple analogy with the real system of heat flow, two parts of high temperature and low temperature are artificially made in a MD system. They can be placed periodically under the periodic boundary conditions [6]. This method can give thermal conductivity, Soret coefficient, etc. This idea can be expanded to the mass flow in the two-component system. Concentration gradient is at the steady state obtained by mass swapping in the concentration controlled regions [7].

1.2.2. Perturbed NEMD In boundary-driven NEMD, steady flow is achieved with steady gradient. Steady flow can be also realized under the constant field using a an additional force to the atoms [8]. An example is ionic flow with a constant electric potential gradient. Steady thermal flow is also obtained. Transport properties are calculated from the linear response theory.

Local Quantities

(Pressure) in Molecular

Dynamics

247

1.2.3. Unsteady state MD simulations at unsteady state seem to become popular because the longer time simulations are possible now. Time development is the target. Various mechanical processes in the atomistic scale such as fractures, dislocations, Martensitic transitions, and so on are in this category. Another examples are formation process of atomic clusters [9] and others and phase transition process.

1.3. Bridge

to macroscopic

systems

Unsteady state process simulations are often done to mimic the real large system; fracturing, sintering, phase separation, and so on. In order to analyze the simulation results, it is necessary to calculate local quantities such as pressure, temperature, concentration, heat flux, etc., because the system is not uniform. In this article, calculation methods of local quantities, particularly local pressure, are discussed. In continuum dynamics, they are well defined. In molecule dynamics, we see atoms and molecules in the system. They are not continuum. These local quantity calculation will become important, when we develop large-scale and multi-scale simulation methods.

2. Local quantities 2.1. Local quantities

in

volume

A quantity per volume, A, in a local volume, Viocai, (See Fig. 1) is easily

O

o y

local

O Fig. 1.

Local quantity in volume Vj o c a ;.

calculated, when the corresponding atomic quantity, a*, of atom i, is given. It is simply a summation of that in volume Viocai as

T.

248

Ikeshoji

A(Vlocal) = - L -

Y,

«*•

(!)

Local density, /?, is, for instance, calculated by this way from atomic mass, rrij, of atom i as PiViocal) = 77 5Z m i ' (2) K oca ' < lev,.,*,, where the summation is for all atoms in the volume, V;oco;. Thermodynamically meaningful ensemble average quantity, A, is a time average of atomic quantity at MD time step t in the volume. A(Vlocal)

= {A(Vlocal))

(3)

1 vlocal(t2-t1

£ { £ *(4

i)step.

+

ep=tl

KiGVlaaal

(4)

)

< • • • > denotes the ensemble average. The local volume is, here, assumed to be constant. Kinetic temperature in V, T(V), can be calculated in this way (in case of three dimensional coordinates):

l = ( Y. m * M < <5) \ZnlocalkB . ^ I where kg is Boltzman constant and niocai is the total number of atoms in the local volume. When there is a net or local flow in the system, the above equation must be modified as

T(Vlocal)

™ -

)

»X<«<-»o)2;,

- \ 5 = £ , £

TO

where VQ is the time-average flow velocity in the volume:

= i ( £ m^

v

0 = T7 (

m

^

ivi )

(7)

\i&Vloc.at

M

= £ step=ti

£ ieViooai

m

*

(8

)

Local Quantities

2.2. Local quantities

(Pressure) in Molecular

Dynamics

249

at point

It is mathematically possible to define a local quantity at a point, R, using a delta function, 6, as all

A(R) = Y/aiS(fl~R),

(9)

i

in which the delta function has dimension of length"3. It is zero except at x=0, but integration of it in the whole space is equal to the unity as <5(r) = 0, f y rt) 8{f)dr = 1.

(10) (11)

J —(

Density at R can be written with this definition as all

p(R) = '£imi6(fi-R).

(12)

i

Integration of this equation gives local density in the volume as: p(Vlocal)

(13)

= fp{R)dr

(14)

-l

= T7 *local

/-OO

Yl

/

/*00

/

/»00

/

m

iti(xi

- x)5(yi - y)S(zi - z)dxdydz

(15)

• J —oo ./— oo J — oo

(16) The summation is for all atoms in the volume. This equation is, of course, equivalent to Eq. (2) and it is practically used in the MD calculations. It is impossible to use Eq. (9) including the delta function directly 3. Local pressure Pressure is defined mechanically as a force onto a surface or thermodynamically as energy necessary to change a volume. From this definition, it is impossible to think a pressure of a single atom. So, it seems to be difficult to define a local pressure at a point. But, it was already shown by Irving and

250

T. Ikeshoji

Kirkwood [10] to be possible as described here. In this section, two similar terms, pressure tensor P and stress tensor a, are used. Since pressure is a thermodynamic quantity and stress is a mechanical quantity, the former is defined as a time-average value and the latter can be instantaneously calculated. They are both the tensor and in the following relationship. (17)

»

In this section, we consider mainly the system consisting of particles interacting by the pariwise potential. 3.1. Local pressure

as a force onto a surface

When a finite size of surface is considered as shown Fig. 2, pressure onto this surface, i.e. a sum of force fa between atoms i and j onto this plane, looks to be denned as force per surface as

p(«)

E £ \i£left j£right

J ij

(18)

ocal

where the direction is shown by superscript ^a\ But, it is arbitrary, since we don't know how the force acts onto the surface. It may be straight, or may be curved to avoid the surface as shown in Fig. 2. If we consider the infinitely large surface, pressure on the surface can be calculated, since the force must pass through the surface. This is known as method of plane (MOP), which is proposed by Todd and Evans [16] to calculate a local pressure on infinitely large surface.

Fig. 2.

Local pressure on a finite surface.

Local Quantities

(Pressure) in Molecular

Dynamics

3.2. Pressure in volume under the periodic conditions

251

boundary

Before moving to the local pressure at point, pressure calculation in ordinary MD of particles with a pairwise potential shall be reviewed. Pressure tensor, P , of N atom system in a unit cell of Lx, Ly, Lz SLX) in size under the periodic boundary conditions is calculated from the virial theorem

\i€cell

I

= PkBT{l}-^/j2

\iGcell j^i

£fa®&})

\ȣce/J j^i

I

(2°)

I

(21)

where

Uj =fj-fi

(22)

and

or nj Symbol means outer product of two vectors to give a tensor. The force vector here defined as a force onto atom i as usual. The distance vector r^ is, however, used to be defined by two ways, In this article, it is defined from i to j . In some papers and textbooks, the opposite definition, from j to i, is used. This virial theorem can be derived as a force onto a surface. Stress in direction a on an infinite surface can be calculated as a sum of the forces passing through the surface S at x as illustrated in Fig. 3

252

T. Ikeshoji

Fig. 3. Local pressure on a surface in the unit cell under the periodic boundary conditions. (Contribution from two atoms is shown.)

where the summation is for all atoms located at the left of the plane and those at the right, and the force coming from the momentum change in the a direction when an atom passes the surface

l1

£

(a)

(25)

2m,- v.

Tt~s x i<x, x i+wj '''6t>x 3

where the summation is for all atoms passing through the surface within time St. The latter force is due to the momentum change of the left part because of disappear of the atom and that of the right; it corresponds to the force due to hitting the wall. The total stress a at time t in the unit cell is obtained by integration of the stress at a; as

1 1

(ax)

Lx Jo

E

It's Xi<X,Xi+v\

X

M"4EE(§

(a)

•dx.

&t>X

(26)

Integration of the first term is valid only the distance where atoms move within St. Integration of the second term is valid for the distance from atoms i to j . Thus:

Local Quantities

(Pressure) in Molecular

Dynamics

-(ax) = v fes («i-}«) 2^(Q) - 5 E E W ^{E^M-SEEW}

253

(27)

(28)

Since pressure is the time average of —a, virial theorem Eq. (21) is obtained. To get Eq. (21), the fact that velocities in the three directions are independent at equilibrium is used for the first term. Non-orthogonal parts in the first term becomes zero from this fact.

3.3. Pressure

in plane

layer

An example of practical uses of the local pressure is to calculate pressure profile through various interfaces. We shall consider the system which is divided into several plane layers parallel to the interface as shown in Fig. 4.

!

k»

t

Fig. 4. Layers divided to calculate pressure profile through a plane interface parallel to the y — z plane.

Pressure in an infinite layer or a layer under the periodic boundary conditions can be obtained by integrating the pressure on surface, Eq. (24) and Eq. (25) in the layer, from x\ to X2 instead of 0 to Lx used in Eq. (26) to obtain the pressure of the whole system.

254

T. Ikeshoji

-*S2r 1

(29)

^

Xi<X,Xi+V;rtidt>X

Xi<X

Xj>X

\

(32) where Ag^ is the distance of part of rij which located in the layer from x\ to X2- Integration part is illustrated in Fig. 5 by arrows. Summation of

Fig. 5. Integration part of straight connecting line between atoms.

the second term is not only for particles in the layer but also for particles of which distance vector passes the layer. Summation of the first term is only for particles in the layer. Pressure profile obtained by this method is shown in Fig. 6. The dip of the pressure is due to the surface tension at the liquid-gas interface. 3.3.1. Incorrect equation of local pressure in layer If the summation of the second term in Eq. (32) is taken only for atoms in the layer and those interacting with them, the correct pressure is not obtained. From this incorrect idea, i.e. a simple extension of the virial theorem in Eq. (21), may give the following equation for the stress.

Local Quantities

(Pressure) in Molecular

255

Dynamics 1.2

0.02 a. 0

•*-*-•*-•••-.»

:•#•-•-*•«•

J;-«-*:^-»

*•

? - • " • • • »-«-^.-«~a

• • • • * - • • - • * -

b) : 0.8

'- 0.4

-0.02

c

0.04 0.02 I

a) :

H.

'B'"R

\ "

0

%

" "

-e

\

-0.02

\

-0.04

~xx •

• - • y y _ j x

•

ZZ

i„i.

-0.06

H . . . .

-0.08

10

15

x

25

20

i

. . . .

30

35

Fig. 6. Pressure profile through a liquid-gas interface of Lennard-Jones particle system. Density profile is shown by a solid line in the upper panel. (Units are in Lennard-Jones reduced units.)

_a(<xx) [incorrect]

1 S(x2

~x{)

^2

2miV

i

EE i^layer

(33)

j

This equation does not give the correct pressure as shown in Fig. 7, in which the pressure profile perpendicular to the layer is wavy, though there is the mechanical balance. The reason of the incorrect results is that Eq. (33) is not obtained by the correct integration. It does not have, for instance, contribution from atom pair both of which are not located in the layer. Another examples are shown in Fig. 8 and Fig. 9, which show the pressure profile through the solid-solid interface of Lennard-Jones crystals (fee) with the different lattice directions. Correct way of Eq. (32) gives the flat profile of the pressure tenor in the normal (xx) direction (Fig. 8). The pressure tensor in the transverse direction (yy and zz) does not show the flat profile because of the non-uniform density when it is calculated in atomic scale. This is understood from a kind of surface tension. When the incorrect equation Eq. (7) is used, the pressure profile in the normal direction is not

T. Ikeshoji

256

0.02

•

0

- » • » - • . - ,

,,,,,,,,, |

1.2

,. , . |, ., . » — -jr--*-

b) : 0.8

i

^ 0.4

c

• ;

0.02

*• NIl^H

....!....

k i * i » i * . ti-*r±

"jn

0.04 . . . . . . . . . . . . 0.02

0

a

-0.02

XX

•-•yy x

-0.04

zz

a'

-0.06 -0.08

0

....

" {V

0

*••*

0

10

15

20

25

30

35

Fig. 7. Incorrect pressure profile through a liquid-gas interface of Lennard-Jones particle system by Eq. (33). (Units are in Lennard-Jones reduced units.)

flat (Fig. 9); it contains a wavy profile in the atomic scale, since atoms are fixed with a vibration.

Fig. 8. Pressure profile through a solid-solid interface of Lennard-Jones particle system by Eq. (32). (Units are in Lennard-Jones reduced units.)

More details discussion on the integration path and about spherical

Local Quantities

(Pressure) in Molecular

Dynamics

257

Fig. 9. Incorrect pressure profile through a solid-solid interface of Lennard-Jones particle system by Eq. (33). (Units are in Lennard-Jones reduced units.)

layers will be discussed in the following section.

3.4. Pressure

at point

Local pressure in an infinite layer seems to be well expressed. But, it is necessary to discuss on the integration path more precisely. Such a discussion was presented by Irving and Kirkwood in 1950 (before molecular dynamics being popular) [10] to give the atomic scale expression for the thermodynamic and kinetic quantities. Schofield and Henderson discussed the pressure in more details [13] along with Irving Kirkwood's work. In this subsection, we describe pressure at point according to Schofield and Henderson's way. In order to get the pressure at a point, momentum density J{R) at point R is considered, at first, It can be expressed using the 5 function as

J(R) = Y,Pi5[R-r] Pi =

rriiVi.

(34) (35)

Time evolution of the momentum density in the a-direction is, then, expressed with path integral along arbitrary path Coi from the origin to r* as

258

T. Ikeshoji

djM(R) dt l

i

v,") XH) (a) (0) Pi p;

I

(36)

p

*

•'C'oi

where $(ri) is potential energy at r*j and 2 is point on the path. Time evolution of momentum density in volume V by continuum mechanics is

dJ(a) v

()

(/v^)

(37) =v dt R When we compare the above two equations, Eqs. (36) and (37), inside of Vg may be equal each other without an integration constant. We get, JR

then, the expression for the a/3 component in the stress tensor;

«-^ f

n (a) D (/3)

(38)

rrii

JCoi

I

where (To is the integration constant. With virial theorem of Eq. (21), the integration constant is 0, and kinetic and conngurational terms may be as follows. Kinetic part (ideal part): (39) Configurational part (non-ideal part): R-l

dl^

(40)

For the two-body potential, it becomes with an arbitrary integration path Cij from atoms i to j .

^ ( 5 ) = | Ei j^kE / if>?ws R-l

dl^

(41)

Ci

In order to use these equations defined at point in the MD calculations, it is necessary to integrate these equations in the volume concerned.

Local Quantities

(Pressure) in Molecular


I

Dynamics

259


(42)

<#»<«-,)— £ ££-•

(*•)

''local JViocal

Kinetic part is simply integrated as

Configurational part becomes

ZVlocal *-?"$£

JCiieVtBcal)

3.4.1. Integration path There is a long history about the discussion on the integration path appeared in Eq. 41. It was emphasized that the integration path is arbitrary [13]. Irving and Kirkwood presented naturally a straight line as an example (see Fig. 10) [10]. It is called now Irving and Kirkwood choice (IK-contour). Harasima showed another path which consists of two parts, a line parallel to a flat or spherical plane and the perpendicular line as shown in Fig. 10 [12]. It is called Harasima choice (H-contour). These two contours have been always raised when local pressure is a subject of papers. But, some questions on the arbitrariness have been presented. Blokhuis and Bedeaux described that H-contour does not obey microscopic sum rule [14]. Wajnryb et al. described that pressure must be independent on numbering of atoms and center of the coordinates [15]. Schofield and Henderson also described that to be symmetry for the pressure tensor the path must be straight line [13]. In order to avoid this kind of arbitrariness another kind of calculation methods of the local pressure were presented. One is Method of Plane (MOP) valid only on infinite pane by Todd and Evans [16]. Lovett, and M. Baus gave " Local thermodynamic pressure " (similar to tangential component in H-contour) [18]. 3.4.2. Pressure tensors in the flat layer by path-integral through IK-contour and H-contour Normal component: In the case of the flat layer under the periodic boundary conditions, the

260

T. Ikeshoji

H-contour Fig. 10.

IK-contour

Irving-Kirkwood (IK) contour and Harasima (H) contour.

pressure tensor by IK-contour is the same as that already discussed in section 3.3. Normal component (xx) of the stress tensor can be written with two coordinates, x\ and X2, which are the left and right side coordinate of the layer. Since integration path is from x^ to Xj, the region which contributes to the integration in the layer can be written from min(x2,Xj) to maix(xi,Xj).

(xx)

E E-^/ Jij

(45)

dx

Xi 5:^2 Xj > X i

A*) (46) Xi<X2 Xj>Xl

=- E E Xil&Xj)

X j *^_Xx

(x) Ax)

r fU'fii' ' ij Jij

V

X2 -

X\

(47)

»

This normal component is independent on the choice of contours. Any contours which connect atoms i and j give the same value as can be understood from Fig. 11. Tangential component: In the case of the tangential component of the stress tensor, the expression depends on the choice of contour. The equation for the IK-contour is similar to that of the normal component as

-o-,

(vv)

„(y) Av) r ij Jij X2 -

V

Jv) ij

Xi

'

(48)

Local Quantities

Fig. 11.

(Pressure) in Molecular

Dynamics

261

A contour to connect atoms i and j .

Integration through the H-contour, contribution is only when atom j is located in the layer.

-4

W)

=-

E

r(y) Av) J M

17 -

(49)

Xa<Xi<Xb

This is the same expression as Eq. (33), which was given as the expression which was simply extended from the virial theorem, but its normal direction component gives the incorrect non-uniform pressure profile through the flat interface at equilibrium. 3.4.3. Pressure in spherical layer In the case of calculation of the pressure in the spherical coordinate, the system is divided into spherical layers as shown in Fig. 12. Pressure in the spherical layers is calculated from Eq. (39) and Eq. (40). Kinetic part does not depend on the choice of the contour. IK-choice in the spherical layer We consider a spherical shell of radius R and thickness AR. For the IK choice of contour dj, the integration path is shown by arrows in Fig. 13. In order to integrate, a new variable A to define the path I is introduced as l = ri + Xfij;

0
(50)

The integration region is in the interval dj £ Vjoca; is defined by Aa < A < At where Aa and Af, are given by a = r\ + Xafij and b = ?i + Xbfij, respectively, and a and b are the entry and exit points, respectively, of dj in the layer. For some configurations of i and j , dj m&y penetrate

T. Ikeshoji

Fig. 12. Layers divided to calculate pressure profile through the spherical interface of a droplet.

Fig. 13.

Integration region by IK-choice in the spherical layer.

the layer twice as seen in Fig. 13, and C^ e Viocai represents two separate intervals. These situations will be understood in the following as the integral fxb actually consisting of two parts (fxb + Jxb'), without being explicitly shown. Furthermore, we assume the notation that Aa = 0 if Fj £ V\0cai and A;, = 0 if fj € Viocai- The configurational part of the aa-component of the stress tensor follows

- 4 * ( Q a ) Wocai) = - £

£

^ fb

(ea • /;-) (eQ • r y ) dX.

(51)

Local Quantities

(Pressure) in Molecular

263

Dynamics

Note that the unit vector, e r , e$, and e^ are not constant through the contour /, when they are expressed in the Cartesian coordinate as (52)

T'T'Tj (ll+l2y)1/2'

lyh

*>TXZ

ee =

m+qyvw+iDw by

e —

Igr

1/2, 2

2

(54)

,0 > .

1/2

~(ll+ly) (l X+l y)

Assuming that fa and f^ are parallel ( ftj = —fij^j/r^), written as IK(rr) -&C Knocal)

fij

f "

<jf^{R) may be

=

2

^ V ^

r%X + 2{fi-rij)r%X

+

{fi-fijf

r?-A2 + 2 ( r i . r y ) A + r?

-^\vlocal) / , / , r~Tr v ,;*

,• 3

Vi,yrij,x

~ ri,xrij,y)

H

I JXa

(53)

i

dX

(55)

= r?M* + 2(rfi,f'fiJ,4>)*

+ rl
dX,

(56) where r i ^ and r ^ are vectors projected of fi and f^, respectively, on the x-y plane. The integrals may be expressed in the closed form, with the results

-a^HVocal)

= ]T £ •>•

r b VI ^

- A«) F^)

- F(X°)]

'

( 57 )

1

where

F

W

- ~\Jririj

~ (^i ' ^j)2

arctan

rfjX +

jri-fjj)

(58)

_\Jririj~(r'i-rij)'2

and lK(4>) m\ _ Jj±_ [G(X ) - G(X )} >(R) b a

njV

(59)

264

T.

Ikeshoji

where

^•» r 'J>

G{\) =

r/ 2 r 2

\l i,4> ij,4> If rjrjj — (ft • fij) center,

r

'.*rai>L-

_ /r - - r

arctan

^2

\ i, ' ij,)

\ri,(j>rij,

~ \ri, ' rij,)

(60) = 0, i.e. particles i and j are on the same line from the

[rfj (Afc - A„) F(Xb) - F(Xa)}

= 1.

(61)

^ rl,rij, ~ (^i<4> ' ~^ih) = ^' *-e- projections of particles z and j on the x-y plane are on the same line from the center, [G(X„) - G(Xa)} = 0.

(62)

In the calculation, the value of A at the crossing point of the connecting line with a sphere of radius R is needed.

Afl = 4 " j " in • rij) ± \l{n • njf

- (rf - B?)\

with 0 < \ R < 1 . (63)

H-choise in the spherical layer: H-contour in the spherical coordinate can be divided into two parts; straight line from atom i toward the center of coordinate and the line on the spherical surface where atom j is located. The former contributes to the radial direction and the latter contributes to tangential direction. Normal component in the stress tensor is very simple. It is similar to the flat layer. TH{rr)

=

i-

\ ^ \ ^

/ f..

( r0rn

2Vlocai For the force in the tangential direction, angle w between atoms i and j along the H-contour is introduced. Then, tangential component becomes

^J.S^MH))-

2V

i€Vloaal

j^t

(65)

Local Quantities

(Pressure) in Molecular

Dynamics

265

3.4.4. Pressure profile in the liquid droplet Pressure tensors in the droplet are calculated using the equations shown in the above. An example of liquid droplets of Lennard-Jones particles is shown in Fig. 14. There is a small but obvious difference between IK and H choices. o.i 0.08 0.06

I""" 1

w®

^/\

D

. ly Har isima u\ zr'n tC ^pSSsggiKajs 0.02 -~i sverse —tran 0 c£ -- By 1 larasinjia -0.02 'm ^

0.04

S

-0.04

Fig. 14.

^

0

2

4

H

6

10

12

14

Pressure tensor in the liquid droplet. Lines are by IK-choice.

Pressure tensor calculation in the liquid droplet have been presented by Thompson et al. [11]. They used both IK- and H-contours to calculate the pressure in the spherical layer and on the spherical plane and found no difference within errors. But, highly accurate calculate in layers showed here the difference. The force balance are satisfied in the both profiles. Which is correct or better? An answer is obtained from the calculation in the homogeneous liquid described in the next part.

3.4.5. Pressure profile in homogeneous liquid with spherical layers Hafskjold and Ikeshoji have recently shown that H-contour does not give uniform pressure for the uniform liquid of hard sphares under the equilibrium when the spherical coordinate is used [19]. When we use the LennardJones particles with equations given in the above section, pressure near the center is different from the bulk value ss shown in Fig. 15, even though the system is uniform, homogeneous, and mechanically balanced. Mechanical balance in the spherical coordinate is expressed as

Pr(r) =PN{r) +

2dpN(r) r dr

(66)

266

T. Ikeshoji

and

f

PN(T)

(67)

pT(r)dr ,

Jo

where superscripts N and T show normal and tangential components. Pressure profile by H-contours shown here satisfies this mechanical balance. When IK-contour is used for the integration, the pressure is uniform shown in Fig. 16. Therefore, the IK-contour seems to be better integration path even for the inhomogeneous fluid.

sure Tensor

1.5

iK n f1

•*WM

i

—

m %£

0.5

•

/

i

0 0

2

4

6

8

0

10

2

4

6

Fig. 15. Pressure tensor by H-contour in homogeneous Lennard-Jones liquid. (The line shown by the arrow is the pressure expected from the force balance.)

1.5

1

i

h

I Mv 1

m

0.5

\ \ 0

Fig. 16.

i

2

4

6

8

10

0

2

4

6

8

10

Pressure tensor by IK-contour in homogeneous Lennard-Jones liquid.

4. Concluding remarks Local quantities are used to bridge the molecular dynamics calculation and macroscopic one. Some quantities are easily bridged but some quantities

Local Quantities

(Pressure) in Molecular

Dynamics

267

of a function of more than two atoms are not so simple. Pressure tensor is such a quantity and discussed here. Integration path problem was not solved completely. But, we found an evidence to avoid Harasima choice to be used to calculated the local pressure tensor. The recommended calculation method of the local pressure tensor in a local volume is to integrate the force along with straight line connecting the two atoms within the local volume as illustrated in Fig. 17.

Fig. 17.

Recommended path integration for the local pressure tensor.

Acknowledgments The work was partially done while the author was visiting the Institute for Mathematical Sciences, National University of Singapore and Institute of High Performance Computing (IHPC) in 2003. The visit was supported by the Institute and IHPC. This work was conducted as a collaboration with professor Bj0rn Hafskjold in Norwegian University of Science and Technology. References 1. S. Nose, Molec. Phys., 5 2 , 255 (1984). S. Nose, Prog. Theor. Phys Suppl, 1 0 3 1 (1991). 2. M. Parrinelo a n d A. R a h m a n , Phys. Rev. Lett, 4 5 , 1196 (1980). 3. M. Sprik a n d G. Ciccotti, J. Chem. Phys., 1 0 9 , 7737 (19 98) a n d references therein. 4. Gear, Kevrekidis C o m p . C h e m . E n g . (2002), A. Laio, M. Parrinello, P N A S 99 (2002). 5. T . Ikeshoji a n d B . Hafskjold, Mol. Phys., 8 1 , 251 (1994). 6. B . Hafskjold a n d T . Ikeshoji, Mol. Sim., 1 6 , 139 (1996). 7. M . J . Gillan, J. Phys. C: Solid State Phys., 2 0 , 521 (1987); D. M a c G o w a n a n d D . J . E v a n s , Phys. Rev., A 3 4 , 2133 (1986). 8. T . Ikeshoji, B . Hafskjold, Y. Hashi and Y. Kawazoe, Phys. Rev. Lett., 7 6 , 1792 (1996).

268

T. Ikeshoji

9. J.H. Irving and J.G. Kirkwood, J. Chem. Phys., 18, 817 (1950). 10. S.M. Thompson, K.E. Gubbins, J.P.R.B. Walton, R.A.R. Chantry, and J.S. Rowlinson, J. Chem. Phys., 8 1 , 530 (1984). 11. A. Harasima, Adv. Chem. Phys., 1, 203 (1958). 12. P. Schofield and J.R. Henderson, Proc. R. Soc. Lond., A379, 231-246 (1982). 13. E.M. Blokhuis and D. Bedeaux, J. Chem. Phys., 97, 3576 (1992). 14. E. Wajnryb, A.R. Altenberger, and J.S. Dahler, J. Chem. Phys., 103, 9782 (1995). 15. B.D. Todd, D.J. Evans, Phys. Rev. E, 52, 1672 (1995). 16. F. Varnik, J. Baschnagel, and K. Binder, J. Chem. Phys., 113 444 4(2000). 17. H.El Bardoumi,, R. Lovett, and M. Baus, J. Chem. Phys., 113, 9804 (2000). 18. B. Hafskjold and T. Ikeshoji, Phys. Rev. E, 66, 011203 (2002). 19. T. Ikeshoji and B. Hafskjold, Molec. Simul, 29, 101 (2003).

RECENT ADVANCES IN MODELING AND SIMULATION OF HIGH-SPEED INTERCONNECTS

M i c h e l N a k h l a and R a m A c h a r Department of Electronics, Carleton University, Ottawa, ON - K1S5B6, Canada E-mail: {msn, [email protected] The rapid increase in operating speeds, density and complexity of modern integrated circuits has made interconnect analysis a requirement for all state-of-the-art circuit simulators. Interconnect effects such as ringing, signal delay, distortion and crosstalk can severely degrade signal integrity. Interconnections can be from various levels of design hierarchy. As the frequency of operation increases, the interconnect lengths become a significant fraction of the operating wavelength, and conventional lumped models become inadequate in describing the interconnect performance and transmission line models become necessary. This chapter describes some of the recent advances in transmission line simulation techniques. Application of mode-order-reduction algorithms to high-speed interconnect analysis is also presented. 1.

Introduction

With the rapid developments in VLSI technology, design and CAD methodologies, at both the chip and package level, the operating frequencies are fast reaching the vicinity of GHz and switching times are getting to the sub-nano second levels. The ever increasing quest for high-speed applications is placing higher demands on interconnect performance and highlighted the previously negligible effects of interconnects such as ringing, signal delay, distortion, reflections and crosstalk. Interconnects can exist at various levels of design hierarchy such as on-chip, packaging structures, multichip modules, printed circuit boards and backplanes. In addition, the trend in the VLSI industry towards miniature designs, low power consumption and increased integration of analog circuits with digital blocks has further complicated the issue of signal

269

270

M. Nakhla & R. Achar

integrity analysis. It is predicted that interconnects will be responsible for majority of signal degradation in high-speed systems 1 " 25 . Highspeed interconnect problems are not always handled appropriately by the conventional circuit simulators, such as SPICE . If not considered during the design stage, these interconnect effects can cause logic glitches which render a fabricated digital circuit inoperable, or they can distort an analog signal such that it fails to meet specifications. Since extra iterations in the design cycle are costly, accurate prediction of these effects is a necessity in high-speed designs. Hence it becomes extremely important for designers to simulate the entire design along with interconnect subcircuits as efficiently Oft 1 ^S

as possible while retaining the accuracy of simulation . Speaking on a broader perspective, a "high-speed interconnect" is the one in which the time taken by the propagating signal to travel between its end points can't be neglected. An obvious factor which influences this definition is the physical extent of the interconnect, the longer the interconnect, more time the signal takes to travel between its end points. Smoothness of signal propagation suffers once the line becomes long enough for signal's rise/fall times to roughly match its propagation time through the line. Then the interconnect electrically isolates the driver from the receivers, which no longer function directly as loads to the driver. Instead, within the time of signal's transition between its high and low voltage levels, the impedance of interconnect becomes the load for the driver and also the input impedance to the receivers 1>15 . This leads to various transmission line effects, such as reflections, overshoot, undershoot, crosstalk and modeling of these needs the blending of EM and circuit theory. Alternatively, the term 'high-speed' can be defined in terms of the frequency content of the signal. At low frequencies an ordinary wire, in other words, an interconnect, will effectively short two connected circuits. However, this is not the case at higher frequencies. The same wire, which is so effective at lower frequencies for connection purposes, has too much inductive/ capacitive effects to function as a short at higher frequencies. Faster clock speeds and sharper slew rates tend to add more and more high-frequency contents. An important criterion used for classifying interconnects is the electrical length of an interconnect. An interconnect is considered to be "electrically short", if at the highest operating frequency of interest, the interconnect length is physically shorter than approximately one-tenth of the

Recent Advances in Modeling and Simulation

wave-length (i.e., length

of High-Speed Interconnects

of the interconnect/X

271

~ 0.1 , A, = v / / ) . E l s e

the interconnect is referred as "electrically long"1'15. In most digital applications, the desired highest operating frequency (which corresponds to the minimum wavelength) of interest is governed by the rise/fall time of the of the propagating signal. For example, the energy spectrum of a trapezoidal pulse is spread over an infinite frequency range, however, most of the signal energy is concentrated near the low frequency region and decreases rapidly with increase in frequency. Hence ignoring the high-frequency components of the spectrum above a maximum frequency, fmax , will not seriously alter the overall signal shape. Consequently, for all practical purposes, the width of the spectrum can be assumed to be finite. In other words, the signal energy of interest is assumed to be contained in the major lobes of the spectrum and the relationship between desired fmax with the tr is the rise/fall time of the signal can be expressed as ' '

'

'

fmax"0.35/tr

(1)

This implies that, for example, for a rise time of 0.1ns, the maximum of frequency of interest is approximately 3GHz or the minimum wave-length of interest is lOcms. In some cases the limit can be more conservatively set as / J max

90

=1/' r

In summary, the primary factors which influence the decision that, "whether high-speed signal distortion effects should be considered", are interconnect length, cross-sectional dimensions, signal slew rate and the clock-speed. Other factors which also should be considered are logic levels, dielectric material and conductor resistance. Electrically short interconnects can be represented by lumped models where as electrically long interconnects need distributed or full-wave models. 1.1

High-Speed

Interconnect

Models

Depending on the operating frequency, signal rise times and nature of the structure, the interconnects can be modeled as lumped, distributed (frequency independent/dependent RLCG parameters, lossy, coupled) or full-wave models.

272

1.1.1

M. Nakhla & R. Achar

Lumped

Models

At lower frequencies, the interconnect circuits could be modelled using lumped RC or RLC circuit models. RC circuit responses are monotonic in nature. However, in order to account for ringing in signal waveforms, RLC circuit models may be required. Usually lumped interconnect circuits extracted from layouts contain large number of nodes which make the simulation highly CPU intensive. 1.1.2

Distributed

Transmission Line Models

At relatively higher signal-speeds, electrical length of interconnects becomes a significant fraction of the operating wavelength, giving rise to signal distorting effects that do not exist at lower frequencies. Consequently, the conventional lumped impedance interconnect models become inadequate and transmission line models based on quasi-TEM assumptions are needed. The TEM (Transverse Electromagnetic Mode) approximation represents the ideal case, where both E and H fields are perpendicular to the direction of propagation and it is valid under the condition that the line cross-section is much smaller than the wavelength. However, the inhomogeneities in practical wiring configurations, give rise to E or H fields in the direction of propagation. If the line cross-section or the extent of these nonuniformities remain a small fraction of the wavelength in the frequency range of interest, the solution to Maxwell's equations are given by the so called quasi-TEM modes and are characterized by distributed R, L, C, G per unit length parameters . In practical situations, owing to complex interconnect geometries and varying cross-sectional areas, the interconnects may need to be modelled as nonuniform lines. In this case, the per unit length parameters are functions of distance 1.1.3

" . Distributed Models with Frequency-Dependent

Parameters

At low frequencies, the current in a conductor is distributed uniformly through out its cross section. However, as the operating frequency increases, the current distribution gets uneven and it starts getting concentrated more and more near the surface or edges of the conductor. This phenomenon can

Recent Advances in Modeling and Simulation

of High-Speed Interconnects

273

be categorized as follows: skin, edge and proximity effects ' ' ' " . The skin effect causes the current to concentrate in a thin layer near the conductor surface and this reduces the effective cross-section available for signal propagation. This leads to increase in the resistance to signal propagation and other related effects . The edge effect causes the current to concentrate near the sharp edges of the conductor. The proximity effect causes the current to concentrate in the sections of ground plane that are close to the signal conductor. To account for these effects, modelling based on frequency-dependent p.u.l. parameters may be necessary. Rest of the chapter is organized as follows. Section-2 provides a detailed analysis of transmission line equations and derivation of a generic multiconductor transmission line stamp, suitable for inclusion in a MNA analysis. In Section-3, a review of formulation of circuit equations in the presence of distributed elements and limitations of conventional simulators is given. Review of efficient techniques for discretization of Telegrapher's equations is given in Section-4. Sections 5-7 give a detailed account of simulation of interconnects using model-reduction techniques. Section-8 provides references to related topics.

2. Distributed Transmission Line Equations Transmission line characteristics are in general described by Telegrapher's equations. Consider the transmission line system shown in Fig. la. Telegrapher's equations for such a structure can be derived by discretizing the line into infinitesimal sections of length Ax and assuming uniform perunit length (p.u.l.) parameters of resistance (/?), inductance (L), conductance (G) and capacitance ( Q . Each section then includes a resistance RAx, inductance LAx, conductance GAx and capacitance CAx (Fig. lb). Using Kirchoff's current and voltage laws, one can write 15 , ~\ v(x + Ax, t) = v(x, t) - RAxi(x, t) - LAx^-i(x, t) at

(2)

or v(x + Ax,£-v(x,t)

=

_R.{X)

{)_^_

t)

(3)

274

M. Nakhla & R. Achar

Taking the limit Ax —» 0 , one gets Av(x, t) = -Ri(x, t) - lJLi(x, t) ox at

(4)

ifO. t)

i(x, t)

i(d, t)

v(U, t)

vtx, t)

v{d,t)

ground, x =0

x x + Ax

x =d

(a) Transmission line system

i(x, t)

uwc—WV

v(x, t)

LAx

x ! *-

R&x CAx Ax

• > , (x + A x )

(b) Representaion of a discretized section Fig. 1. Transmission line system and discretization Similarly, we can obtain the second transmission line equation in the form: 3-('(x, t) - -Gv(x, t) - C^-v(x, t) ox at

(5)

Taking Laplace transform of equations (4) and (5) one can write d V(x, s) = -(R + sL)l(x, s) = -ZI(x, s) ox

(6)

I(x,s) = -(G + sC)V(x, s) = -YV(x,s)

(7)

dx

Recent Advances in Modeling and Simulation

of High-Speed Interconnects

275

where Z and Y represent the p.u.l. impedance and admittances of the transmission line, given by Z=R + sL; Y = G + sC (8) The set of equations represented by (6) and (7) can be solved if they can be written in terms of one of the unknowns (either V(x, s) or I(x, s) ) as follows -ZjVix, s) = ZYV(x, s) = TZV(x, s) dx -^jl(x,s) dx

= YZl{x,s) = I I(x,s)

(9)

(10)

where Y(s) is the complex propagation constant, given by T(s) = CC+./P = JZY = J(R+jwL)(G+jwC)

(11)

where a represents the real part of the propagation constant and is known as attenuation constant, whose units are expressed in (Nepers/m). p represents the imaginary part of the propagation constant and is known as phase constant, whose units are expressed in (radians/m). Solution of (9) and (10) is given as a combination of forward-reflected waves travelling on the line as V(x,s) = V(0,s)e±Yis)x

(12)

±r{s)x

I(x,s) = l(0,s)e

(13)

The phase shift and attenuation experienced by the travelling waves are given by

e

and

ai s x

e-

->

>

respectively. If the lines are lossless, the

propagation constant is given by, Y(s) = ;'P = JZY = jwjLC in this case represents a pure-delay element. 2.1

Multiconductor

Transmission

Line

. The line

System

Consider the multiconductor transmission line (MTL) system, with /V coupled conductors, shown in Fig. 2.

M. Nakhla & R. Achar

vx{x,t)

v,(0, t)

vi(d,t)

i2(d, t)

i2(x, t)

i2(0, t)

v2{x, t)

v 2 (0, t)

v2(d, t)

iN(d, t)

iN(x, t)

iN(0, t) 1

vjv(0, t)

i\{d,t)

ix(x,t)

»i(o,0

•

vN(x, t)

vN{d, t)

£

ground . x =0

Fig. 2. Multiconductor transmission line system Using the steps similar to the case of single transmission line, we can derive the multiconductor transmission line equations. Per-unit-length parameters (R, L, G & C) in this case become matrices and voltage/current variables become vectors represented by v and i, respectively. Noting these changes, we can re-write (4) and (5) as v(x,t) dx i(x, t\

0 G

R v(x, t) 0 i(x, t)_

0 C

L 0

v(x, t) i{x, t)

Taking Laplace transform of equation (14) one can write

(14) (15)

Recent Advances in Modeling and Simulation

l-V(x, ox p(x,s) ox

S)

of High-Speed Interconnects

277

= -ZI(x, s)

(16)

= -YV(x,s)

(17)

where Z and Y represent the impedance and admittance matrices, given by Z = R + sL,

Y = G + sC

(18)

The R, L, G & C matrices are obtained by a two-dimensional solution of Maxwell's equations at appropriate positions, along the propagation axis. For this purpose, depending on the nature and geometry of the structure, and the desired accuracy, techniques based on quasi-static or full-wave approaches can be used. The R, L, G & C matrices are symmetric and positive definite 2.2

'

.

Multiconductor

Transmission

Line

Stamp

In this section, we derive a stamp relating the terminal currents and voltages of MTL structures, suitable for inclusion in SPICE-like simulators. The transmission line stamp equations.

is derived, through decoupling

of

MTL

From (16) and (17), we get two sets of coupled wave equations as a2 -^V(x,s)=ZYV(x,3) dx -5L/(* ( ,) = YZI(x, s) dx

(19)

(20)

Decoupling of equations in (19) or (20) can be achieved through the use of suitable modal transformation matrices . For this purpose, introduce a transformation W relating the circuit voltages V and modal voltages V as V(x,s) = WV(x,s)

(21)

278

M. Nakhla

& R.

Achar

Hence (19) can be re-written as (for simplicity, we omit the accompanying term (x,s)) — V = (W ZYW)V dx

(22)

For effective de-coupling of equations to take place, the matrix product in parenthesis must lead to a diagonal matrix as,

W lZYW

\]

0 0

0

0

(23)

0 0 Y2 where the diagonal matrix contains the eigenvalues of the product which corresponds to the roots of the characteristic equation |Y^£/-Zr| = 0;

k=l,2,...,N

ZY, (24)

where U represents the unity matrix (we assume the general case that, there exist N distinct eigenvalues). Having obtained the propagation constants, solution of (22) can be written in the standard form as Vk(x) = e where

Vk(x)

cki + eK ckr;

k = 1,2

(25)

N

represents the k? modal voltage and cki,

ckr

corresponding constants, pertaining to incident and reflected respectively. Eq. (25) can be written in the matrix form as Vx(x) V2(x)

VN(x)

-v

are the waves,

-\r -Y„x

'2r

(26)

-Ni

L

Wr

Recent Advances in Modeling and Simulation

Define, E{x) - diag

of High-Speed Interconnects

279

- V and pre-multiplying both sides of (26) .. e by the modal transformation matrix W (from (21)), we can write (26) in terms of circuit voltages as -V

V(x) = W[E(x)]Cl + W[E(x)]

-l.

(27)

C2

where Cl and C 2 are constant vectors, which can be determined from the terminal currents and voltages (i.e. at x = 0 and x = d). A relationship between the near-end (x=0) and far-end (x=d) voltages can be derived using (27) as V(0) V{d)_

w

w

WE(d)

W[E(d)]~

(28)

Next, substituting (27) in (16), we have *i

-WT[E(x)]C1 + WT[E(x)]

(29)

C2 = -ZI(x); YA

or -i.

Wj = z

I(x) = W / [ £ W ] C 1 - W / [ £ ( x ) ] 'C2

wr

(30)

A relationship between the near-end (x-0) and far-end (x=d) can be derived using (30) as 7(0) I{d)\

W. WtE(d)

-w; -l

(31)

-Wt[E(d)]

Using (28) and (31) and eliminating the constants, Cx and C 2 , we get the representation in terms of y-parameters as

M. Nakhla & R. Achar

280

1(0) I(d\

-W;

w. WtE(d)

-W^d)}-

w

w

WE(d)

W[E(d)]~

V(0) V(d)_

(32)

Assume that the multiconductor stamp is required in the standard form shown in Fig. 3. In this case, to account for the current 1(d) as flowing inwards, the expression for 1(d) in (32) must be multiplied b y - 1 . Noting this and simplifying (32) further, we can write the MTL stamp in terms of yparameters as

W(0) =

7(0)

[V(d)_

Yl2

V(0)

^22_

V(d)

Y~u _^21

(33)

where (-Y t d)

{-rkd)

(Tkd)

;

E, = diag

E2 = diag<

-e

(Ykd)

(k= 1,2, ...,N) (34)

I(d, s)

/(0,5) V(0,s)-

MTL

System

V(d, s)

Fig. 3. Multiconductor transmission line system 2.2.1 Matrix Exponential Stamp An alternative form of the MTL stamp is also quite popular and it has the matrix exponential form , which is explained below. Equations (6) and (7) can be written in the hybrid form as

Recent Advances in Modeling and Simulation

dx

V(x, s) = (D + sE) V(x, s) ; I(X, 5)_ _I(x, s)_

D=

of High-Speed Interconnects

0 -R ; -G 0

E=

0 -L -C 0

281

(35)

Solution of (35) can be written as (D + sE)d V(0, s) V(d, s) =e I(d, s) W *).

(36)

A relationship between the forms represented by (33) and (36) can be obtained as follows: Define T(s) as ^11

T(s)

r

i2

(D + sE)d

(37)

^21 ^22

Using some algebraic manipulations, we can express the relationships between the hybrid parameters (36) and the y-parameters (33) as -Tn'Tu

7(0) I(d\

V(d) I(d)_

/

i

21~ 22*i2

M2 Y

2i

+

Y

Y22 u

V(0) V(d\

12 Ml

MI

22*12

12 -1

MI

~YKYX2

V(0) 7(0).

(38)

(39)

Similarly, another useful representation of the MTL stamp is in terms of ABCD parameters, which can be written as

V n -u

V(0) +

hi 7

0" 7(0)

22 -U I(d\

=

0 0

(40)

In the next section, we will review a generic formulation of distributed interconnect circuit equations, suitable for general purpose circuit simulators.

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3.

Formulation of Circuit Equations

The MNA and output equations for lumped linear networks can be written using a generic notation as Wx(t) + Gx(t) = Bu(t) y =

L

u{t)

^

where B and L are selector matrices, with entries (0 or 1). The superscript T ' denotes the transpose. Let b(t) = Bu{t). From (41), MNA equations in the frequency-domain can be written as (G + sW)X(s) = b(s) Y(s) = LTU(s)

(42)

For the case of nonlinear elements, MNA equations in (41) can be modified as Wx(t) + Gx(t) + F(x(t))-b(t)

=0

y = LTu(t)

(43)

where F(x(t)) is a nonlinear function of x. 3.1

Formulation Distributed

of linear subnetworks Elements

containing

Consider a linear subnetwork n containing distributed elements. Using (33), the frequency-domain equations of a distributed subnetwork containing nd coupled conductors can be written as Yd(s)Vd(s) = Id(s)

(44)

where Vd(s) and Id(s) represent the Laplace-domain terminal voltages and currents of the distributed element, respectively, Yd(s)

represents the

admittance matrix having complex dependency on frequency, which are

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described in terms of line parameters. Equation (42) representing the lumped linear network can be combined with (44) as ^

n +

Gn Ld

(45)

YL dl where • Wm G„ G 91 K

n

are constant matrices describing the lumped memory

and memoryless elements of subnetwork n, respectively, 91 n is the node-space of subnetwork n, • Ld is the selector matrix which maps the terminal currents of the distributed subnetwork to the nodal space of the linear subnetwork % . • b% e 91 is a constant vector with entries determined by independent voltage/current sources of subnetwork n, Vn(s) e 91 is the vector of node voltage waveforms appended by independent voltage source currents, linear inductor current waveforms of linear subnetwork JI . The equation (45) can be concisely written as

(46)

W(s)X(s) = b(s)

3.2

Generic formulation of nonlinear circuits with distributed elements

Consider a general network containing arbitrary number of nonlinear and linear (lumped and distributed) components. For simplicity, let the linear components be grouped into a single linear subnetwork n as shown in Fig. 4. Using (43), without loss of generality, the circuit equations network 0 can be written as % ^ ' )

+ G

^W

+ t

A W + ^ ( 0 ) - » t ( 0 = 0,

139

te[0,T]

for the (47)

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where W,xN.

•

W. G e SR

are constant matrices describing the lumped mem-

ory and memoryless elements of network 0 , respectively, 4.G5R

is a

constant vector with entries determined by the independent voltage and current source, • F(xJ is a function describing the nonlinear elements of the circuit, N

xJt) € 91 * is the vector of node voltage waveforms appended by independent voltage source current, linear inductor current, nonlinear capacitor charge and nonlinear inductor flux waveforms, AL is the total number of variables in the MNA formulation and nn is the total number ports for linear subnetwork TI . • Ln = [/,. •] with elements ie {1, ...,N±},j€

{1, ...,«„}

/, e { 0 , 1}

where

with a maximum of one nonzero in /i

each row or column, is a selector matrix that maps in(t) e 91 the vector of currents entering the linear subnetwork n, into the node space 91^* of the network § , The linear multi-terminal subnetwork n can be characterized in the frequency-domain by its terminal behavior as Yn(s)VK(s) = IK(s) (48) where

Yn(s)

is the y-parameter matrix of subnetwork n,

Vn(s)

is the

vector of terminal voltage nodes that connect the subnetwork to the network <)>, / (s) is the Laplace transform (in(t)).

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Linear subnetwork n

Vi

*ik n

-•

• > = '

^ n l '•••• .71. •

.

^n-nJt.-.TE.

Fig. 4. Nonlinear network (|) containing linear subnetwork 71 with distributed elements 3.3

Interconnect

simulation

issues

Simulation of large interconnect networks is associated with two major bottlenecks: Mixed frequency/Time problem and CPU expense. 3.3.1

Mixed frequency/Time

Problem

The major difficulty in simulating high-frequency models such as distributed transmission lines is due to the fact that, while described in terms of partial differential equations, they are best represented in the frequency-domain (48). As seen, they do not have a direct representation in the time-domain. On the other hand, nonlinear devices can only be described in the timedomain (47). These simultaneous formulations are difficult to handle by a traditional ordinary differential equation solver such as SPICE 2 6 ' 1 3 9 " 1 5 0 .

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3.3.2

M. Nakhla & R. Achar

CPU Expense

Frequency-domain simulation of large linear networks is conventionally done by solving (42) or (45) at each frequency point using LU decomposition and forward-backward substitution. For time-domain simulation, integration techniques are used to convert a set of time-domain differential equations into a set of difference equations. For example, application of trapezoidal rule to (43), leads to a nonlinear set of difference equation (G + T-W)v(t + At) + F(v(t + At)) = (j-W-GJv(t)

+ b(t) + b(t + At) -Fv(t) (49)

To solve (49) at each time point, Newton iterations are required, which may need several LU decompositions. This causes (note that W and G matrices for interconnect networks are usually very large) the CPU cost of a timedomain analysis to be expensive. The objectives of interconnect simulation algorithms are to address both mixed frequency/time problem as well as to handle large linear circuits without too much of CPU expense. There have been several algorithms proposed for this purpose, which are broadly classified into two main categories, as follows, (a) Approaches based on macromodelling each individual transmission line set. Techniques such as "method of characteristics" are grouped in this category and are discussed in Section 4. (b) Approaches based on model-order reduction (such as AWE, CFH, PRIMA) of the entire linear subnetwork containing lumped as well as distributed subnetworks and are discussed in Sections 5-7. It is to be noted that the second approach can also be used in conjunction with the first approach.

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4. Simulation Techniques Based on Transmission Line Macromodels In this approach, transmission-line networks described by Telegrapher's equations (partial differential equations) are translated into a set of ordinary differential equations (known as the macromodel), through some kind of discretization. The conventional approach ' for discrete modelling of distributed interconnects is to divide the line into segments of length Ax, chosen to be small fraction of the wavelength. If each of these segments (assume that the line is discretized into 'M' segments) are electrically small at the frequencies of interest (i.e. Ax = L/M « X), then each segment can be replaced by lumped models. Generally lumped structures used to discretize MTL equations contain, the series elements : L(x)Ax and R(x)Ax, and shunt elements: G(x)Ax and C(x)Ax. (L(x), R(x), G(x), C(x) are the per unit length inductance, resistance, conductance and capacitance of the line, respectively (Fig. 1).

4.1

Distributed v/s Lumped: Number of Lumped Segments

It is often of practical interest to know how many lumped segments are required to reasonably approximate a distributed model. For the purpose of illustration, consider LC segments, which can be viewed as low pass filters. For a reasonable approximation, this filter must pass at least some multiples of the highest frequency fmax of the propagating signal (say ten times, / 0 > 10/ max ). In order to relate these ' , we make use of the 3-db pass band of the LC filter given by

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where d is the length of the line. From (1), we have fmax = 0.35/r r using (50), we can express the relation f0^10fmax

and

in terms of the delay of

the line and the rise time as — , > 10 x 0.35/r r , or •Kid

r

tr>3.5(nxd) = \0Td

(51)

In other words, delay allowed per segment is 0.1 r r . Hence the number of segments (AO is given by N = (\0xd)/tr

(52)

In the case of RLC segments, in addition to satisfying (51), the series resistance of each segment must also be accounted. The series resistance Rd representing the ohmic drop should not lead to impedance mismatch which can result in excessive reflection within the segment 2 ' 3 . However, one of the major drawbacks of the above conventional discretization is that it requires large number of sections, especially for circuits with high-operating speeds and sharper rise times. This leads to large circuit sizes and the simulation becomes CPU inefficient. In order to overcome these difficulties, several techniques for efficient discretization are proposed in the literature. These methods can be broadly classified, based on the passivity property as follows. (1) Macromodels with no guarantee of passivity: a sample of such techniques is the Method of Characteristics (2) Macromodels with guaranteed passivity by construction: a sample of such techniques are Integrated Congruent Transform and Exponential Pade based Matrix-Rational Approximation.

4.2

Method of Characteristics

The method of characteristics (MC) transforms partial differential equations of a transmission line into ordinary differential equations containing time-delayed controlled sources.

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Consider the case of two conductor transmission lines, as shown in Fig. 5a. An analytical solution, in terms of y-parameters for (6) or (7) can be derived 45

as 1

/ = YV,

z0(i-e-2^)

(53) -2e~Vd

1 + e-2^d

V-

where y is the propagation constant on the line, and Z 0 is the characteristic impedance. V-y and /j are the terminal voltage and current at the near end of the line, V2 and l2 are the terminal voltage and current at the far end of the line. The y-parameters of the transmission line, are complex functions of s, and in most cases cannot be directly transformed into an ordinary differential equation in the time domain. The MC succeeded in doing such a transformation, but only for lossless transmission lines. Although this method was originally developed in the time domain using what was referred to as characteristic curves (hence the name), a short alternative derivation in the frequency domain will be presented here. By re-arranging the terms in (53) we can write, Vj = ZQ/J + g-Yd[2V2 - e-yd(Z0Ix + Vj)]

(54)

V2 = Z 0 / 2 + e-Jld[2V1 - e~yd(Z0I2 + V2)] Next, (54) can be re-written as (55)

v2-z0i2 = w2 where

wx = e-yd[2v2-e-yd(z0il W2 =

e-Y

d

+ vl)]

(56)

[2V1-e-^(Z0/2+y2)]

Using (54) and (56), a recursive relation for Wj and W2 can be obtained as

290

M. Nakhla & R. Achar

w, W2 =

e-vd[2V? - WU (57)

e-yd[2Vl-Wl]

A lumped model of the transmission line can then be deduced from (54) and (57), as in Fig. 5b.

hit)

Ut)

Vj(0

vM) x = d

(a)

(b)

Fig. 5. Macromodel using Method of Characteristics If the lines were lossless (in which case the propagation constant is purely imaginary; y = y'(3), the frequency domain expression (57) can be analytically converted into time-domain using inverse Laplace transform as wx(t + i) 2v2(t)-w2(t) (58) w2(t + x) = 2 v 1 ( 0 - w 1 ( 0 4.3

Exponential

Pade based Matrix-Rational

Approximation

This algorithm directly converts partial differential equations into timedomain macromodels based on Pade rational approximations of exponential 41,42,89 . In this technique coefficients describing the macromodel are matrices computed a priori and analytically, using closed-form Pade approximant of exponential matrices. Since closed-form relations are used, this technique doesn't suffer from the usual ill-conditioning experienced with the direct

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291

application of Pade approximations. Hence it allows higher-order of approximation. Also it guarantees the passivity of the resulting Macromodel. Consider the exponential form of Telegrapher's equations describing the multiconductor transmission lines, given by (36), V(d, s) = e I(d, s)_

D

V(0, s)

Z = (D + sE)d;

7(0, s)_

0 -R -G 0

E=

0 -L -C 0

(59)

2 where d is the length of the line. The matrix e is approximated using matrix-rational function as {Z)e^QNM{Z) (60) N,M where PN M(Z) and QN M(Z) are polynomial matrices expressed in terms of closed-form Pade rational functions as rn.MW

- L(M

(M + N-JVM + N)\j\(N-jy}

j '

i =o M

UN,MW

~ L(M

(M+N-j)\M\ j + N)\j\(M-j)r

j=o

Several recursive relationships exist for PN M(Z) and QN M (Z) such as

(61)

M. Nakhla & R. Achar

292

P

N,M&)

P

N,M(Z)

~

P

N-UM(Z)

+ Z{{M

= PN,M-l(Z)

P

+ N){M

+

+ N)(M

+

N_1J

+Z

l(M

N-I,M-I(Z)

P

N_1y)

N-l,M-l(Z) (62)

QN,M(Z)

= QN- 1, « ( Z ) + Z{(M

2/v, M ( Z ) = 2/V,M-l( Z ) + Z ^ M

+ N^M

+

/ v - 1 )J QN~ 1. M- 1 ( Z )

+ A f ^ M + A ,_ 1 ^]2A'-l,A/-l(

The fact that the coefficients PNM(Z)

Z

)

and g w M (Z) are known a priori in

closed-form, provides substantial computational advantage for the proposed algorithm. The Pade rational function of (61) for M = N = n can be represented in terms of subsections, obtained by the pole-zero pairs as o =

[Pn,n(Z)]~lQnyn(Z)

=

n/2

T] l(aiU-Z)(ai*U-Z)]'i[(aiU

+ Z)(a*U + Z)](63)

i = 1

for even values of n, and

^, n ( z )]"'e n ,„( z ) 9 = (n-l)/2 _1

[a 0 f/-Z] [a 0 t/ + Z]

Y\

[(a,.f7-Z)(a 1 *t/-Z)r 1 [(a,t/ + Z)(a,*f/ + Z)] (64)

i = l

for odd values of n, where U represents the unity matrix, ai = xt +jyt are the complex roots for (' > 0 and a0 is a real root. The symbol

represents the

complex conjugate operation. It is to be noted that the PN M(Z)

is a strict Hurwitz

polynomial.

This means

coefficients

a{ and aQ in (7) and (8) are positive

definite

The hybrid stencil for such an i subsection can be written as

polynomial that

the

constants.

Recent Advances in Modeling and Simulation

[Pn,n(Z)],-+,

Vi+1 ',

+

of High-Speed Interconnects

lQn.«Wl

293

(65)

l

Proof of passivity of the macromodel can be found in 4 1 . Also the extension of the closed-form matrix-rational approximation based technique to handle frequency-dependent parameters can be found in . 4.4

Delay

Extraction

Although algorithms based on method of characteristics (MC) provide fast solutions for long low loss lines, they do not guarantee the passivity of resulting macromodels. This was substantiated by several numerical tests which showed that the MC can lead to non-passive macromodels. Moreover, it has been widely shown in the literature that, the transient analysis of a nonpassive macromodel with other passive circuit elements may lead to spurious oscillations. On the other hand, algorithms based on matrix rational-function approximations (MRA) guarantee the passivity of macromodels. However, in the presence of large delay lines (e.g. long lines with small losses), this may require high-order approximations (to accurately capture the flat delay portion) leading to inefficient transient simulation. To address the above issues, a new algorithm for passive macromodeling of transmission line subnetworks was recently introduced . It employs a mechanism for delay reduction prior to performing the matrix rational approximation. The new algorithm leads to significantly lower order macromodels for long lossy coupled lines. A brief discussion of the algorithm is given below. Using perturbation and assuming that ||A|| «||i m a x B| (where smax corresponds to the maximum frequency of interest), (36) can be approximated as 160

e(A

+ sB)

- em

UeCk, k=

(Ck = f(A,*))

(66)

1

where IJCJ » |C 2 | » ... ||C m |. It is to be noted that, the Lie product formula 161

M. Nakhla & R. Achar

294

provides a systematic alternative approach to obtain (66) with an error estimation (e ) and is given by (A + sB)

U-tJ \

+ e„

sB

A

m

m

• e

e

lk.NO

;

(67)

*= l

Next, a theorem is introduced which enables more accurate delay extraction by modifying the Lie product formula given in (67). Theorem: The product S

A „

+ £„

A A

2m

Qt = e

m

2m

(68)

e e

* =1 +s

converges asymptotically to e this case is given by max

a s m - ^ « . The associated error (em) in (A+sB)

(69) m Equation (68) henceforth is referred to as Modified Lie Formula - I. If ||A|| « |5 majc B| (which is the case for long low lossy lines), then an alternative form for (68) can be used with a reduced error, and is given by (referred to as Modified Lie Formula - II):

UQ> = 0

0<S<S„

k= 1

ne* k-

i B A £B 2m m 2m • e e e

+ £„

(70)

1

Also it can be proved that, average of the approximations in (68) and (70) as given by (referred to as Modified Lie Formula - III): / A

ne* + e„ k = 1

s

e*

S

JL A

2m Ve

m

e

A. 6.

2m

e

2m m

+e

e e

±B.\ 2m

(71)

J

further reduces the error (note that exact error estimates of each of these formulae are derived, however not given here due to lack of space). Fig. 6. demonstrates numerically, the accuracy comparisons of Lie's formula and proposed approximations (Modified Lie Formula - 1 , II and III), for a typical set of line parameters.

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4.4.1

Frequency-Dependent

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295

Parameters

In the case of frequency-dependent parameters, it is ideal to extract a maximum delay, without affecting the transmission line causality conditions. Let B(s) = B{s)-Bmax. In general a logical choice for Bmax is B(°°). Next, (68), (70) can be modified as: m

sB

A(s)+sB(s)

n e

t

+ em;

Qk^e

max A(s) + sB(s)

e

e

(72)

k= 1 m

n e * + em;

sB

max A(s)+sB(s)

Qk = e

e

sB

max

e

(73)

k= 1

The products represented by (72)or (73) can be viewed as a cascade of m transmission lines. In addition, each of the kth product term can be viewed as a cascade of lossy and lossless transmission lines. The lossy terms are macromodeled using the passive matrix rational approximation. The resulting macromodels are of significant lower orders (since a significant delay portion is already extracted from these terms). They are later combined with the lossless terms using the method of characteristics approach. For example, each Qk in (73) can be realized as shown in Fig. 7 It is to be noted that passivity of the entire macromodel is now guaranteed as the passivity of each sub-line in (73) is preserved. Based on the knowledge of the norm of A and B line parameter matrices and the maximum frequency of interest, approximation represented by (72)or (73) is selected and required order (m) satisfying the pre-defined error tolerance can be determined using (69). It is to be noted that for relatively short lossy lines, delay extraction may reduce the efficiency of the MRA macromodel. Based on the knowledge of line parameters and error-estimates of MRA and modified Lie formulae, a criterion has been developed to select the appropriate macromodel (i.e., MRA or MRA with delay extraction).

M. Nakhla & R. Achar

296

Lie Formula Modified Lie Formula I Modified Lie Formula II Modified Lie Formula III

order (m) Fig. 6. Comparison of Error Estimates of Lie Formula and Modified Lie Formulae -1, II and III.

sB

max 2m

e (Lossless)

A(s) + sB(s) m

e (MRA)

sB

max 2m

e (Lossless)

Fig. 7. Macromodel Realization of the Product Terms in (73).

5. Model-Reduction Based Simulation Algorithms Interconnect networks generally tend to have large number of poles, spread over a wide frequency range. Even though majority of these poles would normally have very little effect on simulation results, however, they make the simulation to be CPU extensive by forcing the simulator to take smaller step sizes. 5.1

Dominant Poles

Dominant poles are in general those, which are close to the imaginary axis and significantly influence the time as well as the frequency characteristics of the system. The moment-matching techniques (MMTs) 77 capitalize on

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the fact that irrespective of the presence of large number of poles in a system, only the dominant poles are sufficient to accurately characterize a given system. A brief mathematical description of the underlying concepts of momentmatching techniques is given below. Consider a single input/single output system and let H(s) be the transfer function. H(s) can be represented in a rational form as H(s) - ^

(74)

where P(s) and Q(s) are polynomials in s . Equivalently, (74) can be written as Np k H(s) = c+ Y —'-

(75)

^ s-Pi i=o

where P • and kt are the i pole-residue pair, N

is the total number system

poles and c is the direct coupling constant. The time-domain impulse response can be computed in a closed form using inverse Laplace transform as h(t) = c8t + £ *,./''

(76)

i=0

In case of large networks, N , the total number of poles can be of the order of thousands. Generating all the N poles will be highly CPU intensive even for a small network and for large networks it is completely impractical. Model-reduction techniques address the above issue by deriving a reducedorder approximation H{s) in terms of dominant poles, instead of trying to compute all the poles of a system. Assuming that only L dominant poles were extracted which give a reasonably good approximation to the original system, equation (74) and the corresponding approximate frequency and time responses can be written as

M. Nakhla & R. Achar

298

L

l

H(s) •* H(s) = %£ = Z + Y —'-*• QW t?0'-Pi

(77)

L

h(t)~h(t)

= c5r+ ]T k/'

(78)

1=0

5.2

Moments of the

response

Consider the Taylor series expansion of a given transfer-function, H(s), at point, 5 = 0, ZJI x

ru s

T„ s

(H(s)){1)

H(s) ~H(s) = H(s) + s^—Yr—

2(H(s)f2) + s

n(H(s))in)

,_m

—9i — + ••• + s *•—*-p—

(79)

where the super-script (n) denotes the nth derivative. Using a simpler notation, we can re-write(79) as " (0 H(s) ~H(s) = m0 + m1(s) + m2s +...+mns = V s mf, mi = —^r~— (80) ( = 0

The coefficients of Taylor series expansion, (w(.) are also identical to the time-domain

moments

of the impulse response h{t). This can be easily

seen by using the inverse Laplace transform of h(t)

,

2 2

st H(s) = jh(t)e~s'dt = \h(t) , l-«+-2J-"

dt

2,

= \h(t)dt + s\{-\)th{t)dt o

o

+ s2\t-^dt+... o

= £ 1=0

s\Q^\tlh(t)dt o

(81)

Due to this analogy, the coefficients of Taylor series expansion, (m,), are generally referred as moments. It has been shown that the moments provide an estimation of delay and rise times 59 ' 60 . Elmore delay 59 , which approximates the mid-point of the

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monotonic step response waveform by the mean of the impulse response, essentially matches the first moment of the response. This can be considered as one of the basic forms of approximation. However, in order to get accurate prediction of interconnect effects, it is essential that the reduced-order model must match (or preserve) as many moments as possible. Several algorithms can be found in the literature for reduction of large interconnect subnetworks . They can be broadly classified into two categories: (1) approaches based on explicitly matching the moments to a reduced-order model, (2) approaches based on implicitly matching the moments. The techniques such as AWE belong to the first category and are discussed in Section 6. Techniques such as PVL, PRIMA, which are based on Krylov subspace formulation, belong to the second category and are discussed in Section 7.

6. Model-reduction based on explicit moment-matching These techniques employ Pade approximation, based on explicit momentmatching to extract the dominant poles and residues of a given system 6 1 " 6 5 . 6.1

Pade

Approximation

Consider a system-transfer function

H(s)

which is approximated by a

rational function H(s) as an + a,s + a1sL+ ... +a,sL Pj(s) 2 H(s) = H(s) = -^ ! - £ - = -fjL \+bxs+...

where a0, ...,aL,bv

...,bM

+ bMsM

(82)

QM(S)

are the unknowns (total of L + M+ 1 variables).

Consider the Taylor series expansion of H(s) at (s = 0 ) , in terms of moments and match it to the rational function approximation given in (82) (hence is the name, moment-matching techniques (MMTs), which is also known as Pade approximation) as follows: an + a,s + a-,sL + ...+a,sL M

1 + bxs + ... + bMs

2

-mjTm^Tmjj

+ m

-r . . . -r ,nn. {L + +

mMsy

L+M

(83)

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Cross multiplying and equating the coefficients of similar powers of 5 starting from s to s on both sides of (83), we can evaluate the denominator polynomial coefficients as m

L-M+

m L-M

m

1

L-M+2

+2 m

•-mL

L+l

m,

b,, 'M

m L+ 1

'M-2

+

m

L+ 1

mL + 2

(84)

mL + M

M-l

The numerator coefficients can be found by equating the remaining powers of s

(from s

to s ) as "0 -

"'0

<2j = m j + b^rriQ

(85) min(L, M) b m

i L-i

1 =1

Equations (84) and (85) yield an approximate transfer function in terms of rational polynomials. Alternatively, an equivalent pole-residue model can be found as follows. Poles pt are obtained by applying a root-solving algorithm on denominator polynomial Q(s). In order to obtain kt, the approximate transfer function given by (77) is expanded using Maclaurin series as H(s) = c- £ s n = 0

•n + 1

V; = oP,-

Comparing H(s) from equations (80) and (86), we note that

(86)

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mn-cj = 0

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301

Pi

(87)

(0 < i < 2L)

•oPi

Residues can be evaluated by writing the equations in (87) in a matrix form as —1 Pl —2

Pl

—1

Pl —1 Pl

—1 PL

-1

--2 PL

0

M0

(88) M

—

L-

—

L-1

Pl

—

PL

L-1

0

L-1 M,

In the above equations c represents the direct coupling between input and output. There are more exact ways to compute c 6.2

Computation

of

Moments

Having outlined the concept of MMTs, we need to evaluate the moments of the system, which are required by (84) - (88). Consider the simple case of lumped circuits and the corresponding MNA equations represented by (42). Expanding the vector X(s) using Taylor series, we have [G+SW] M0 + Mts + M2s +

H

(89)

where Mi represents the i moment-vector. Equating coefficients of similar powers of s on both sides of (89) we obtain the following relationships

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M. Nakhla & R. Achar

GM0 = ,

(90) ;/ > 0

GM, = -WM.1 - 1

Above equations give a closed form relationship for the computation of moments. The moments of a particular output of interest (which are represented by mt in equations (82) - (88), are picked from moment-vectors M; . As seen, (90) requires only one LU decomposition and few forward/ backward substitutions during the recursive computation of higher order moments. Since the major cost involved in linear circuit simulation is due to LU decomposition, MMTs yield very high speed advantage (100 to 1000 times) compared to conventional simulators. 6.3

Generalized

Computation

of

Moments

In the case of networks containing transmission lines, moment-computation is not straight-forward. A generalized relation for recursive computation of higher-order moments can be derived as follows ' ' ' 7 1 ' 7 4 . Consider (46), expanding \\i(s) and X(s) in Taylor series, at an expansion point s = a, we have74 V v|/(a) + -

(«)|

0)1

1!

M0 + Ml(s-a)

where \|/

V

- ( s - a ) + ... + •

denotes the n

+ ...+

Mn(s-a)

-{s-a)

H

(91)

derivative of \|/(.y), and Mn denotes the nth

moment of X(s) at 5 = a . Equating coefficients of similar powers of s = a on both sides of (91), we have WM0 = b WMn

= - £ r= 1

(V (r) )M„. (92)

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Generalizing (92), a recursive relation for any n be obtained as

303

higher-order moment can

[\y(a)]M0 = b (93)

" ( V I )Mn r= 1

It can be seen that the coefficient on the left hand side of (93) does not change during higher-order moment computation. Hence it requires only one LU decomposition and n forward-backward substitutions to compute n moments. Also it is easy to note that, the lumped networks are a special case of (93) (where \|/ r ) = 0 for r>2 in which case (93) reduces to the form given by (90)). Relation (93) requires the derivatives of (\\i). These can be obtained using (45) as

'wn w

W

0

=

0 ;

yA

[V]

(r)

0 (r>2)

=

?A

0

(94)

0

Using (45), transmission line moments can be computed as The derivatives Tp can be obtained as a function of the derivatives of the entries on the RHS of (38) and proper application of Leibnitz's theorem. However, this requires the derivatives of the exponential stamp represented (37). A brief review of computation of these derivatives 6.4

Transmission

Line

is given below.

Moments

Consider the exponential stamp represented by (37). We wish to expand the exponential matrix in Taylor series, as follows: e

(D + sE)d

„

„

= Fn0 + F,s +

+ Fnsn

(95)

From the property of matrix exponentiation of an arbitrary matrix A , we have

304

M. Nakhla & R. Achar A 1 A A A' e = 1 + 1! — +2!— +... +n\•

(96)

A = (D + sE)d

(97)

Let

Hence (96) can be re-written as (D + sE)d _ 6

1 ;

(D + sE)d 1!

|

({D + sE)df 2!

f

'"

+

|

((D + sE)d)n n!

(98)

Expanding the RHS of (98) further, and collecting the terms in powers of s, we have I

(D + sE)d

Dd

1 „2 ,2

+ j | (£>£ + £Z>)/ + | ; (D 2 £ + Z>£D + ED2) d3 +

^

^ £ 2 d 2 + ^{D3E + EDE2 + E2DE + E3D)d3 + . (99) Equating (95) with (99) gives „ / Dd 1„2.2 F o = ^ + -f7 + 2 l Z ) d + -

=f

„ o,o

+ /?

„ „ o,i+fo,2 +

Fj = ^ + j-(DE + ED)d2 + ^(D2E + DED + ED2)d3 + . = FL0

+

F

l.l+Fl,2+-

F2 = ^E2d2 + ^(D3E + EDE2 + E2DE + E3D)d3 + ... F

2, 0 + F2, 1

+ F

2, 2 + '

(100)

and so on. From the above results, a recursive relationship for generating transmission line moments can be obtained as

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F

i = X FiJ>

(DFii_,+EFl_li)d

F

uj =

J

J

i+j

of High-Speed Interconnects

:

305

<* <W *°> (' +» * 0

J=o

FUJ = 0

(i<0 o r ; < 0 ) ;

F0>0 = /

(1Q1)

Convergence of (101), in practice requires 20-30 terms. It is to be noted that the convergence of the series represented by (96) can suffer, if for the first few terms, An grows quicker than n!. In order to control this problem, note that the growth of An depends on its eigenvalues. If all the eigenvalues of A are within the unit circle in the complex plane, then An will decay with increasing n, leading to fast convergence. From (96) one can see that, the eigenvalues of A can be controlled by varying the length d. By restricting d to be small enough, such that the eigenvalues of (D + sE)d will also be small (over a given frequency range), so as not to cause truncation errors or slow convergence. This can be achieved efficiently, by noting that gP + sE)d\ nD + sE)d\ i 2 ) K 2 J

(D + sE)d

e

=e

e

(102)

In other words, moments of a line can be generated by squaring half-line moments. Let <6 represent the half-line moments, then (D + sE)d

e

_

„

n

' = F0 + F1s+ ... +Fns

= (O 0 + OjS + ... +
(103)

which will give,

(=0

The line can be subdivided by power of 2 (i.e, 2 sections, four sections, 8 sections...) and the moments of the smallest section that meets the convergence requirements are calculated. From these, the moments of the entire line can be recursively calculated with the help of (104).

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6.5

Limitations of Single Expansion MMT Algorithms

Obtaining a lower-order approximation of the network transfer function using a single Pade expansion is commonly referred as Asymptotic Waveform Evaluation (AWE) in the literature. However, due to the inherent limitations of Pade approximants, MMTs based on single expansion often give inaccurate results. The following is a list of those properties which have the most impact on MMTs. • The matrix in (84) (which is known as Toeplitz matrix) becomes increasingly ill-conditioned as its size increases. This implies that one can only expect to detect 6 to 8 accurate poles from a single expansion. • Pade often produces unstable poles on the right hand side of the complex plane. • Pade accuracy deteriorates as we move away from the expansion point. • Pade provides no estimates for error bounds. In addition, there is no guarantee that the reduced-model obtained as above is passive. Passivity implies that a network cannot generate more energy than it absorbs, and no passive termination of the network will cause the system to o r Q'j

go unstable . The loss of passivity can be a serious problem because transient simulations of reduced networks may encounter artificial oscillations. In systems containing distributed elements the number of dominant poles will be significantly higher, and it is very difficult to capture all of them with a single Pade expansion. This lead to the development of multi-point expansion techniques such as complex frequency hopping (CFH), which are summarized in the next section. 6.6

Complex Frequency Hopping

CFH extends the process of moment Matching to multiple expansion points (hops) in the complex plane near or on the imaginary axis using a binary search algorithm . With a minimized number of frequency point expansions, enough information is obtained to enable the generation of an approximate transfer function that matches the original function up to a predefined highest frequency of interest. Using the information from all the expansion points, CFH extracts a dominant pole set as illustrated in Fig.

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307

8(b). In addition, CFH provides an error criterion for the selection of accurate poles and transfer functions. 6.6.1

Selection and Minimization of Hops in CFH

A Pade approximation is accurate only near the point of expansion and its accuracy decreases as we move away from the point of expansion (hop). In order to validate the accuracy of such an approximation, at least two expansion points are necessary. Accuracies of these two expansions can be verified by matching the poles generated at these two hops (referred as pole-matching based approach). Alternatively, the two hops can be verified for their accuracy by comparing the value of the transfer functions due to both these hops at a point intermediate to them (referred as transferfunction based approach) . CFH relies on a binary search algorithm to determine the expansion points and to minimize the number of expansions. The steps involved in the binary search algorithm for both the above approaches are summarized below.

Radius of convergence of Pade expansion at .s=0

Real (a) Dominant poles from AWE

(b) Dominant poles from CFH Fig. 8. Illustration of CFH

308

6.6.2

M. Nakhla & R. Achar

Transfer

Function

Based

Approach

In this approach the transfer functions obtained at various hops (expansions) are used to ensure the accuracy of the reduced-order model up to the highest frequency of interest. Steps involved in the algorithm are given in Fig. 9. Note that the computational effort needed for a comparison as required by Step 5 is trivial as the responses can be computed in a closed-form using the transfer functions generated in Steps 2 & 3. Here e[h is a per-defined threshold relative error in the transfer functions. At the completion of the binary search algorithm, a set of transfer functions are generated. When evaluating the frequency response at a frequency point a, only the transfer function which is valid in the region containing a is used. This is repeated for all other frequency points to obtain the frequency response of the system. 6.6.3

Pole-Matching

Based

Approach

In this approach poles of the transfer function are explicitly evaluated at each hop and the hops are verified for their accuracy by comparing the poles from two adjacent hops using a binary search algorithm. If a matching pole is found between two adjacent expansions, then the binary search is stopped. The distance between the matching pole and the expansion point under consideration defines the radius of accuracy for the corresponding expansion. All the poles which are within the radius of accuracy are treated as accurate poles and are retained in the final pole-set. The poles which are outside the radius of accuracy are considered as inaccurate poles and are discarded. Once a set of dominant poles are obtained, residues of the system are obtained using (88). Further details of CFH and its search algorithms can be found in 7 4 and

75

.

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Step 1:

Set fL

= 0 and

fH

= fmax

Step 2:

Expand system's response at frequency

309

. fL

= 0.

coefficients of the corresponding transfer function

Determine

H^(s)

the

using (84)

and (85). Step 3:

Expand system's response at

fjj

= fmax

cients of the corresponding transfer function 'Step 4:

Set

fmid

= -(fL

Hfjij^fmitj)

+fR)

.

Calculate

using the transfer function

• Determine the coeffiHH(s) . HL(j2llfmid) coefficients

and obtained in

Steps 2 and 3. -Step 5:

If

\HH(j2nfmid)

expand at fm^ Step 6:

- HL(j2nfmid)\ and obtain Hmid(s)

< e,fc , Go To Step 6. Otherwise .

If the threshold condition specified by step 5 is satisfied, STOP. ELSE repeat steps 2-5 between every two consecutive frequency points (e.g., between fL & fmid and fmid & fH ) .

Fig. 9. Transfer function based binary search algorithm

7.

Model-Reduction Based on Krylov-subspace Techniques

The direct moment-matching techniques such as AWE have some disadvantages associated with them. First one among them is the illconditioning associated with the moment-matrix. Due to this difficulty, the number of good poles that could be extracted from any expansion point is generally fewer than 10 poles. The second major difficulty is that, they do not guarantee the passivity of reduced-order models. In order to address these difficulties, a parallel class of algorithms, which can be classified as indirect moment-matching techniques were developed 7 9 - 9 °.

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310

These algorithms are based on what is known as Krylov-subspace formulation and Congruent transformation. One of the main features of these algorithms is that they construct the reduced-model based on the extraction of leading eigenvalues (those with the largest magnitude) of a given system (on the contrary, the reduced models from the CFH technique is based on extracting the dominant poles of a given system). In the rest of this section, we will describe the concept and important features of these algorithms. 7.1

Preliminaries

Recall from Section-3, the time-domain MNA and the corresponding output equations can be represented in the form: Cx{t) + Gx{t) = Bu(t);

C,Ge9\nXn;

w = L x(t);

Le SR"X

5e9tnxl

xe9inxl / 1Q5 X

where n represents the total number of MNA variables. Pre-multiplying both-sides of (105) by G , we can write Ax(t) = x(t)-Ru(t);

A = -G~lC,

w = LTx(t)

R = G"' B (106)

Taking the Laplace transform of (106), we can write sAX(s) = X(s)-RU(s) W(s) = LTX(s)

(107)

Rearranging (107), we can write the transfer function Y(s) of the given system as Y(s) = ^=LT(I-sA)-lR U(s) where / is an identity matrix.

(108)

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7.1.1 Why direct Pade based approximation conditioned?

(moment-matrix)

311

is ill-

Consider the transfer-function of a system, as represented by (108). Expanding it in terms of Taylor series, we have Y(s) = LT(I + sA + s2A2 + 53A3 + ... + sqAq)R 1

sk(LTAkR)

= £ k = 0 1

k

= V s mk *=°

T

where

k

mk = L A R (109)

Ideally, increasing the order-of the Pade approximation (which is equivalent to matching more number of moments), should have given us better approximation results. However, in practice, this is true only up to very limited order, beyond which Pade approximation will not yield any better results 7 9 , 1 5 9 . This can be explained by examining the nature of higher-order moments, which are given by mk = L A R.

As can be seen, when

successive moments are explicitly calculated, they are obtained as powers of A. With the increasing values of '&', this process quickly converges to an eigenvector corresponding to an eigenvalue of A with the largest magnitude. As a result, for relatively large values of '&', the explicitly calculated moments mk,mk+x,mk + 2, ..., will not add any extra information to the moment-matrix, as all of them contain information only about the largest eigenvalue. In other words, the rows beyond '£' of moment-matrix are almost identical (or parallel to each other) making the matrix illconditioned. 7.1.2

Relationship

between eigenvalues and poles of the system

In this section we will show the correspondence between the leading eigenvalues and poles of the system. It is important to understand this concept as the Krylov-subspace based techniques obtain the reduced-models

312

M. Nakhla & R. Achar

by extracting the leading eigenvalues of a given system. Consider (106), and assume that the matrix A can be diagonalized in the form .,-1

(HO)

FXF

where X = diag\Xx X2 ... XA is a diagonal matrix, whose diagonal elements represent the eigenvalues of matrix A. The matrix F contains the eigenvectors of matrix A . Using (110), the transfer-function represented by (108) can be re-written as Y(s) = LT(I-sFXF

*) 1R

= LTF(I-sX)

\F *R

1 \-sXl = LTF

F 1R 1

1-^J

[

(HI) which can be simplified as *i Y(s) = £1—sXj

(n/X,.)

=• ^s5 - ( l / X ) J

(

k,

(112)

= 1; *-'s-p

i

where n ( is a function of eigenvectors of matrix A , kt represent the residues. From (112), we can draw following inferences: (a) poles pt are the reciprocal of eigenvalues of matrix A ; the leading eigenvalues (those with largest magnitudes) correspond to the poles closer to the origin, (b) the transfer function of Y(s) can be easily obtained in terms of poles and residues, once the eigenvalues and eigenvectors of A are available. However, for large interconnect circuits, it would be impractical to compute all the eigenvalues and eigenvectors. Hence in the following sections, we will review some of the efficient techniques to extract leading eigenvalues.

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7.2

Computation

of Eigenvalues

of High-Speed Interconnects

of Matrix

313

'A'

In general, the numerical computation of all the eigenvalues and eigenvectors of a given matrix A becomes exceedingly expensive as its size gets above few hundreds. The general approach in such cases is to approximate A with a smaller matrix A , such that the eigenvalues of A axe reasonable approximation of the leading eigenvalues of A . Due to the relatively small size of A, finding its eigenvalues will be a much simpler problem, than finding the eigenvalues of A . Next, we will review some of the basic matrix forms , which would be helpful in understanding the eigenvalue computation algorithms presented in this section. Upper Hessenberg Matrix: A matrix H is called Upper Hessenberg if Hy = 0

for

( ( > ; ' + 1 ) . For example, consider an upper Hessenberg

matrix of order q, having the following form (which is known as companion form)

H

0 1

0 0

0 0

0

1 0

0

. . 0 . . 0 -c .

(113)

. 0 . 1 One of the important advantages of the above companion form is that, its characteristic polynomial, p(x), can be analytically computed and is given by
PM = X ctx

i-i

(114)

i = l

The roots of p(x) give the eigenvalues of H. Orthogonal Matrices: A real square matrix Q is orthogonal if Q Essentially this implies QQT = QTQ = i

=Q • (115)

M. Nakhla & R. Achar

314

All columns, qi (or rows) of orthogonal matrices have unit two norms or INI 2 =

1

( w m c n i m P n e s that qt q{ = 1) and are orthogonal to one another

(which means that q{ q, = 0). QR decomposition: Let AT be a mxn matrix with m>n . Suppose that K has full column rank. Then there exists unique m x n orthogonal matrix Q and a unique upper triangular matrix R with positive diagonals (ru > 0) such that K = QR. There are several techniques (such as modified GramSchmidt orthogonalization process) available in the literature, for this purpose

159

.

Next, consider the circuit equations (106), and a simple similarity transformation as follows AK = KHq (116) where the transformation matrix K is defined as K

=[R

and H

has the upper Hessenberg

Obviously, since H

(11?)

AR ... A'*] companion

form

discussed above.

is related to the matrix A through a similarity

transformation, its eigenvalues are the same as that of A . Although it looks straight-forward, this approach has the following limitations: Computation of H

using the relation (116) ( Hq = K~ AK ) requires the

inverse of the matrix K. However, AT is a dense matrix and hence computation of its inverse will be expensive. Also, K is likely to be illconditioned since the columns of K are formed based on the sequence AlR , which as shown in Section 7.1, quickly converges to the eigenvector corresponding to the largest eigenvalue. In the next section, we will describe general techniques to overcome these problems. These algorithms belong to a class of methods known as Krylov-subspace techniques.

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7.3

of High-Speed Interconnects

315

Krylov-Subspace methods for Iterative Computation of Eigenvalues

We will start by replacing the matrix A'in (116) with an orthogonal matrix Q such that for all q, the leading q columns of K and Q span the same space. This space is called a Krylov subspace, and is denoted by K{A,R, q). In other words, any vector which is a linear combination of the leading q columns of K can be expressed also as a linear combination of the leading q columns of Q. Mathematically we will express this as K(A, R, q) = ColumnSpace ( L

^R

AqR\

= ColumnSpace[Q]

HIS")

In contrast to matrix K, the matrix Q has the following advantages: • Q is well conditioned, —1

T

• It is easy to invert since Q = Q , • Most importantly, we can compute only as many leading columns of Q as needed to get accurate solution (more details about this is covered later in this section). The next question is, how do we get the matrix Ql This can be achieved, by performing QR decomposition on matrix K. Writing K = QR, we can modify (116) as Hq = K~lAK R = =

{QR^AiQR) (R-'Q^AiQR)

QTAQ = RHR'1

=H

i

(119)

Since R and R~ are both upper triangular and Hq is upper Hessenberg, it is easy to prove that the new matrix,

H = RHqR~

, is also upper

Hessenberg. The implications of (119) is that we can reduce the matrix A of

M. Nakhla & R. Achar

316

dimension nxn to a smaller upper Hessenberg matrix H of dimension q x q using orthogonal transformation. In addition, the eigenvalues of smaller system H are approximations of the first q leading eigenvalues of larger system represented by A. Next, we will show that the columns of Q can be computed one at a time giving us the advantage of computing only as many leading columns of Q as needed. One of the popular approaches used for partial reduction of a large matrix to a smaller upper Hessenberg matrix by computing Q, is known in the literature as Arnoldi's given in the next section. 7.4

algorithmi2~^9'159.

Arnoldi Algorithm for (partial)

Assume Q = \ql

q2

More details about this is

Reduction

... q^\ , where q( represents the i column of matrix

Q. From (119) we have AQ = QH

(120)

Recall that all columns, q{ (or rows) of orthogonal matrices have ||#J

= 1

(which implies that q{ qi - 1) and are orthogonal to one another (which means that qi g = 0). Using this information, the first few steps in obtaining the Q and H matrices are outlined below. Since the ||9i||2 = 1 , an easy way to compute it, is to divide the vector R by its magnitude \\R\\2 (we get a unit vector in the direction of R). This step is illustrated in Fig. 10a. qx=R/\\R\\2.

(121)

To determine q2 and the first column of H, we multiply A by the first column of Q. This gives us Aqx, which is the first column on LHS of (120). Equating it with the first column of RHS, we have A 1\ = h\\9\+h2lq2

(122)

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317

Premultiplying both sides by q[ we have qxAqx = qxAqx

hnqlq1+h21qlq2

= hn

(123)

Knowing the value of hn and using the fact that |g 2 | = 1, we can compute h21 from (122) as ^21 = || A *i -^n^ili •

(124)

The direction for q2 can be obtained using (122) as (illustrated in Fig. 10b) Aqi-hnq1 Vi

(125)

'21

Aqx i

1\

I ^21

>«i

\R\

(a)

-^1

> « 1

(b) Fig. 10. Illustration of steps in Arnoldi algorithm

Similarly the rest of the columns of Q and H matrices can be obtained by generalizing the above steps. Note that we didn't need to explicitly compute the product A R . As a result, we were able to avoid the ill-conditioning problem arising due to the quick

M. Nakhla & R. Achar

318

convergence of the sequence \R

AR

A^R

A3R

to the eigenvector of

the largest eigenvalue. The columns qt computed by Arnoldi algorithm are called Arnoldi vectors. The loop over i updating z corresponds to the modified algorithm

Gram-Schmidt

which subtracts the components in the directions q} to g, away

from z, leaving them orthogonal to z- Computing a total of '&' Arnoldi 9

vectors costs k matrix-vector multiplications involving related cost.

A, plus 0{k n)

There are several alternative methods available in the literature for finding the Krylov-subspace . For example one can use multiple passes of orthogonalization, to increase the robustness of the modified Gram-Schmidt orthogonalization process. To recap, we started with the circuit equations

Cx{t) + Gx(t) = b{t)

T

and

—1

w = L x(t). We formed the product A = G C. Using orthogonal transformation, we were able to determine the leading eigenvalues of A which correspond to the dominant poles of the transfer-function. In the following section, we will show how to use this information to perform circuit reduction. 7.5

Circuit Reduction

Using Arnoldi

Algorithm

Finding the reduced-order circuit equations can be explained by a change of variables in (105) by mapping the vector x of dimension n into a smaller vector x of dimension q {q«n) using the orthogonal matrix Q : x = Qx

(126)

Using (126) we can re-write Laplace-domain circuit equations in (106) as sAQX(s) =

QX(s)-RU(s)

W(s) = LTQX(s)

(127)

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of High-Speed Interconnects

T

319 T

Pre-multiplying both sides of (127) by Q and using the relation QQ = I, we have X(s) = (l-sQTAQ)'1QTRU(s) W(s) = LTQ(I-sQTAQ)'lQTRU(s)

{ m )

Hence the transfer-function of the reduced system can be written as Y(s) = LTQ(I-sH)'lQTR

(129)

Comparing the original transfer-function Y(s) represented by (108) with the transfer-function Y(s) of the reduced system represented by (129), we can draw the following conclusions. The eigenvalues of Y(s) are given by the eigenvalues of H. However, since the eigenvalues of H are good approximation of the leading eigenvalues of A, we can conclude that the eigenvalues of the transfer function of the reduced system are good approximation of the poles of the original transfer function. An important criterion during the above reduction is the accuracy of the response of the reduced system given by (129). The frequency response of the reduced system (129) is also a good approximation of the frequencyresponse of the original transfer function (108). An indicator for the accuracy of the response of the reduced system is the total number of moments it can preserve (match), for a given order of reduction (q). It can be proved that the reduced system (129) of order q preserves the first q moments of the original network . In essence, we are able to implicitly match the moments and obtain a reduced-model without the need to directly use the moments as in the AWE algorithm. Hence we will not suffer from the same numerical illconditioning which is associated with direct moment-matching algorithms. The accuracy of the Arnoldi approximation gradually increases as the order q is increased since more moments of the original transfer function will be matched.

M. Nakhla & R. Achar

320

A question that may possibly arise here: how are the accuracies of Arnoldi based approximation and direct Pade based approximation are compared? It was shown in section-8 that a Pade approximation of order q matches the first 2q moments. However, an Arnoldi based reduction of order q matches only first q moments . Essentially, this means that, for a comparable accuracy, the reduced-model from Arnoldi will have double the size of the reduced model from direct Pade based approximation (in other words, direct Pade based models are more optimal). On the other hand, due to the illconditioning, direct Pade based approximation can't achieve higher-order approximation, where as Arnoldi based approximation can.

8. Related Topics and Further Reading In addition to the interconnection simulation algorithms discussed here, there are several related topics which may be of interest to the reader. 8.1

Passivity

Preservation

Passivity implies that a network cannot generate more energy than it absorbs, and no passive termination of the network will make the system unstable. Passivity is an important property, because, stable but not passive macromodels can lead to unstable systems when connected to other passive systems. The loss of passivity can be a serious problem because transient simulations of nonpassive networks may encounter artificial oscillations. This is illustrated in Fig. 11, which represents the transient response of a reduced-order macromodel of a large linear RLC circuit, when connected to an external load of 5QKQ. . Several algorithms were proposed in the literature for preservation of passivity during the reduction of interconnect networks 85 " 9 2 .

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0

0.5

of High-Speed Interconnects

1 Time(seconds)

321

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Fig. 11. Transient response of a nonpassive macromodel with passive terminations 8.2

Full-Wave

Models

At further sub-nano second rise times, the line cross-section or the nonuniformities become a significant fraction of the wavelength and field components in the direction of propagation can no longer be neglected. Consequently, full-wave models which take into account all possible field components and satisfies all boundary conditions are required to give an accurate estimation of high-frequency effects. However, circuit simulation of full-wave models is highly involved. The information that is obtained through a full-wave analysis is in terms of field parameters such as propagation constant, characteristic impedance etc. A circuit simulator requires the information in terms of currents, voltages and circuit impedances. This demands a generalized method to combine modal results into circuit simulators in terms of a full-wave stamps. References ' ' , provide solution techniques and moment generation schemes for such cases.

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Achat

data

In practice, it may not be possible to obtain accurate analytical models for interconnects because of the geometric inhomogeneity and associated discontinuities. To handle such situations modeling techniques based on measured data have been proposed in the literature 12 " 14,100 " 112 . In general, the behavior of high-speed interconnects can easily be represented by measured frequency-dependent scattering parameters or time-domain terminal measurements. However, handling measured data in circuit simulation is a tedious and a computationally expensive process. 8.4

EMI

Subnetworks

Electrically long interconnects function as spurious antennas to pick up emissions from other nearby electronic systems. This makes susceptibility to emissions a major concern to current system designers of high-frequency product. Hence the availability of interconnect simulation tools including the effect of incident fields is becoming an important design requirement. References provide analysis techniques for interconnects subjected to external EM interferences and also for radiation analysis of interconnects. 8.5

Sensitivity

Analysis

Sensitivity analysis involving large interconnect subnetworks can be highly CPU intensive. Model-reduction based approaches provide an efficient means for this purpose 113 " 117 . Acknowledgements The work was partially done while the first author was visiting the Institute for Mathematical Sciences (IMS) and Institute of High Performance Computing (IHPC) in 2003. The visit was supported by the Institute, the National University of Singapore and IHPC.

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C. R. Paul, "A comparison of the Contributions of Common-Mode and Differential -Mode Currents in Radiated Emissions," IEEE Trans. Electromag. Compat., pp. 189-193, May. 89. C. R. Paul, "A SPICE model for multiconductor transmission lines excited by an incident electromagnetic field," IEEE Trans. Electromag. Compat., No. 4, pp. 342-354, Nov. 1994. I. Wuyts and D. De Zutter, "Circuit model for plane-wave incidence on multiconductor transmission lines," IEEE Trans on EMC, vol 36, no. 3, pp. 206212, Aug. 1994. E. S. M. Mok, G. I. Costache, "Skin-effect considerations on transient response of a transmission line excited by an electromagnetic wave," IEEE Trans, on EMC, vol. 34, no. 3, pp. 320-329, Aug. 1992. Y. Kami and R. Sato, "Circuit-concept approach to externally excited transmission lines," IEEE Trans, on EMC, vol 27, no. 4, pp. 177-183, Nov. 1985. N. Ari and W. Blumer, "Analytic formulation of the response of a two-wire transmission line excited by a plane wave," IEEE Transactions on Electromagentic Compatibility, vol. 30, no. 4, pp. 437-448, Nov. 1988. C. R. Paul, "Literal solutions for the time-domain response of a two-conductor transmission line excited by an incident electromagnetic field," IEEE Trans, on EMC, vol. 37, No. 2, pp. 241-251, May 1995. P. Bernardi, R. Cicchetti and C. Pirone, "Transient response of a microstrip line circuit excited by an external electromagnetic source," IEEE Trans, on EMC, vol. 34, No. 2, pp. 100-108, May 1992. G. J. Burke, E. K. Miller, and S. Chakrabarti, "Using model based parameter estimation to increase the efficiency of computing electromagnetic transfer functions," IEEE trans, on Magnetics, vol. 25, No. 4, pp. 2087-2089, July 1989. F. M. Tesche, M. V. Ianoz, T. Karlsson, EMC Analysis Methods and Computational Models, Wiley, New York, 1997. A. K. Agrawal, H. J. Price and S. H. Gurbaxani, "Transient response of multiconductor transmission lines excited by a nonuniform electromagnetic field," IEEE Trans. Electromag. Compat, vol. 22, No. 2, pp. 119-129, May 1980. Y. Kami and R. Sato, "Circuit-concept approach to externally excited transmission lines," IEEE Trans. Electromag. Compat., vol 27, no. 4, pp. 177-183, Nov. 1985. R. Raut, W. J. Steenart and G Costache, "A note on the optimum layout of electronic circuits to minimize electromagnetic field strength," IEEE Trans. Electromag. Compat., pp. 88-89, Feb. 1988. C. W. Ho, A. E. Ruehli and P. A. Brennan, "The modified nodal approach to network analysis," IEEE Trans. Circuits and Systems, vol. CAS-22, pp. 504509, June 1975.

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A. E. Ruehli, Circuit Analysis, Simulation and Design, Noth-Holland, New York: 1988. J. K. White and A. S. Vincentelli, Relaxation Techniques for the simulation of VLSI Circuits, Boston: Kluwer Academic Publishers, 1990. D. O. Pederson, "A historical review of circuit simulation," IEEE Transactions on Circuits and Systems, vol. CAS-31, no. 1, Jan. 1984. A. S. Vincentelli, "Circuit Simulation," in Computer Design Aids for VLSI Circuits, P. Antognetti, D. O. Pederson and H. De Man (editors). Martinus Nijhoff Publishers, 1986, pp. 19-112. J. K. Ousterhout, "CRYSTAL: A timing analyzer for NMOS VLSI Circuits," in Proc. 3rd Caltech. Conf. on VLSI, Mar. 1983, pp. 57-69 Norman P. Jouppi, "Timing analysis and performance improvement of MOS VLSI design," IEEE Transaction on Computer-Aided Design of ICs, vol. 6, no 4, pp. 650-665, July 1987. S. Lin, M. M. Sadowska, and E. S. Kuh, "SWEC: A step wise equivalent conductance timing simulator for CMOS VLSI circuits," in Proc. Electron. Design Automation Conf, 1991, pp. 142-148. A. S. Vincetelli, E. Lelarasmee, and A. Ruehli, "The waveform relaxation method for the time-domain analysis of large scale integrated circuits," IEEE Trans. Computer-Aided Design, vol. l,no. 3, pp. 131-145, Aug. 1982. A. Devgan and R. A. Roher, "Adaptively controlled explicit simulation," IEEE Trans, on Computer-Aided Design, vol. 13, no. 6, Jun. 1994. S. Lele, "Compact finite difference schemes with spectral-like resolution," Journal Compt. Physics, vol. 103, pp. 16-42, 1992. J. Vlach and K. Singhal, Computer Methods for Circuit Analysis and Design, New York: Van Nostrand Reinhold, 1983. T. Kailath, Linear Systems. Toronto: Printce-Hall Inc., 1980. C. T. Chen, Linear system theory and design. New York: Holt, Rinehart and Winston, 1984. L. Weinberg, Network Analysis and Synthesis, NY: McGraw-Hill Book Company Inc., 1962. W. Louis, Network analysis and synthesis. New York, NY: McGraw-Hill, 1962. E. Kuh and R. Rohrer, Theory of Active Linear Networks, San Francisco: Holden-day Inc, 67. E. A. Guillemin, Synthesis of Passive Networks, New York: John Wiley and Sons Inc., 1957. M. E. V Valkenburg, Introduction to Modern Network Synthesis. New York: John Wiley and Sons Inc., 1960. U. S. Pillai, Spectrum Estimation and System Identification. New York: Springer-Verlag, 1993.

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MULTISCALE MODELING OF D E G R A D A T I O N A N D FAILURE OF I N T E R C O N N E C T LINES D R I V E N B Y ELECTROMIGRATION A N D STRESS GRADIENTS Robert Atkinson and Alberto Cuitino Department of Mechanical and Aerospace Engineering Rutgers Piscataway, New Jersey 08854, USA E-mail: [email protected]

University

This article introduces a multiscale modeling approach for simulating a key aspect affecting the reliability in narrow interconnect lines: migration, growth and coalescence of voids driven by electromigration and stress gradients. This approach is based on the Monte Carlo method with a Hamiltonian that contains both short-range and long-range interactions. The short-range interactions are resolved at atomic scale while the contributions of the long-range interaction are computed by solving a subsidiary continuum boundary value problem for the elastic and electric field. Thus, this approach combines both discrete and continuum techniques to provide a numerical tool to follow the complex evolving topologies that are derived from void migration, growth and coalescence. The predicted results are in good agreement with key experimental observations and other numerical and theoretical estimates.

1. I n t r o d u c t i o n T h e research into line interconnect failure has seen many changes in the past 30 years. In the early to mid 1970's, the failure of line interconnects was understood to be primarily due to grain boundary diffusion. [12] [21] This was due to the size of the lines at t h a t time. In fact, it was due to the relative size of the grains in the line, in comparison to the line width. Spawning over a decade, investigations into the mean times of failure ( M T F ) of line interconnects began in the early 1980's.[13] [10] [28] By the early 1990's it had been reported t h a t the M T F of line interconnects level off and begin to increase at some critical line width. These investigations were concerned with the effects of continually decreasing line width on the M T F . T h e results clearly indicated t h a t there was a minimum M T F t h a t occurs

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when the line width is just slightly larger than the grain size. This is due to the transition from one primary mode of failure to another. The transition from grain boundary diffusion to surface diffusion and void motion occurs when the medium for grain boundary diffusion is removed. The removal of the medium occurs when there are no more diffusion pathways along grain boundaries that lay parallel to the line. In fact, the grain boundaries that do exist in small lines, less than 3 fim, are primarily perpendicular to the lines. This gave rise to the term bamboo structure of a line interconnect. It was at this point in the evolution of line interconnects that a new approach toward understanding had to be undertaken. In addition, accompanying experimental research into grain orientation and failure location in the line was completed. [37] [11] These investigations gave understanding into the reasons for failure of certain grain orientations and locations along the lines. A number of simulation strategies have been developed to follow the evolution of interconnect surfaces and internal voids using a continuum approach, where different mechanisms of atomic migration have been considered including electromigration as well as surface and bulk diffusion assisted by stresses [30] [40] [5] [32] [35] [19] [43] [2] [38] [4] [44] [42] [39] [41] [40] [27] [14] [33] [3] [18]. In the present approach, however, we merge discrete and continuum methodologies to describe the mass transport process in interconnects driven by electromigration and stress sources, which is described in the Section 2 and the predictions of the simulations are presented in Section 3. 2. Multiscale Discrete-Continuum Formulation The geometry and topology of the line is defined by indicating the initial position of the atoms, which is a discrete representation of the material. The evolution of the atomic positions are driven a Monte Carlo scheme that considers both discrete and continuum effects. In particular the multiscale bridging link of this methodology is in the formulation of the Hamiltonian, which is defined as follows: Na

Na

EB Si S SiS

H = J2J2~ ( > J) i i=l

j —1

Na

+ Yl\EE^

+E s

°( *)+EA^Si

w

i=l

where, EB (si,Sj) is the pair potential energy between atoms that occupy lattice sites i and j , EE (si) is the electric potential energy at lattice site i, Ea (si) is the elastic strain energy at lattice site i, EA (SJ) is the activation energy at lattice site i, and Sj is the lattice binary counter that

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takes the value of 1 when the lattice site is occupied and 0 when empty. The first term in the Hamiltonian accounts for the short-range interactions discretely, while the second term introduces the long-range interactions via the evaluation of the per site specific energies, which are obtained by surrogate continuum boundary value problems. The geometry and topology utilized for solving of these continuum problems are dictated by discrete simulation. In particular, we consider a Local Monte Carlo approach based on the Kawasaki algorithm. The probabilities for the movement of an atom to an open position are assigned based on the values adopted by the Hamiltonian, Eq.(l) for the before and after configurations. Our approach uses the existing lattice to calculate the solution for the current densities, and elastic energies in the line. The current densities are solved using a finite difference technique while the elastic energies are obtained by a finite element approach. These are continuum solutions to problems that require boundary conditions. These boundaries are determined by the occupation state of the lattice. The solutions to the electric and stress problem are then overlaid on the lattice and each lattice site is then assigned a value based on the calculations from the finite difference and finite element solutions. This is an iterative scheme that requires the continuous update of the electric and elastic fields due to change in the geometry and topological changes driven by the local Monte Carlo selection process. This mixed discrete-continuum methodology, while consistent with continuum solutions, also provides a natural recourse for capturing the motion and aggregation of discrete/point defects leading to interconnect lines damage. The description of the discrete and continuum formulations are presented in Section 2.1 and Section 2.2 respectively. 2.1. Discrete

Formulation

We consider a Local Monte Carlo scheme to follow the evolution of the topology and geometry of the interconnect line, where the transition probabilities are computed based on the Hamiltonian Eq. (1). The equilibrium exchange rates between energy state /i and energy state v is given as follows: P (M -> v) ^ 9 (M -> v) A (M -> v)

(2)

where g is the selection probability of an atom which is always equal to -^-, where Na is the total number of atoms. Therefore the relation simplifies to the following.

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P(fi^v)

Cuitino

A(fi^v)

(3)

Now the determination of the function A must be accomplished. Having identical but distinguishable particles implies that the configuration entropy is given by the following relation: S =

N\ Ni\N2\...

(4)

where N is the total number of energy states, and Ni is the total number of atoms that exist in energy level i. Using the Stirling approximation In (AH) SiNlnN-

N

+-IH(2TTN)

(5)

For large N, In (AH) S AT In AT- N.

(6)

Thus,

In (S) = £ N^ In ( £ NA - £ > „ In (A^) JV =

(7)

E i V ^ Q E d J V " = 0 ' £ = E £ " =*pY,E*dN* = ° (8)

where fi denotes all energy states. Constraints in differential form are zero, so adding those terms to din (5) is acceptable. dln(S)

£ d dNIn (5)

a + PE^ dN„

(9)

IL

where d In (5)

5

cWM

cW^

5>„ln K> M - J ^ l n ^ )

= In AT-In A^. (10)

Thus, at a maximum dln(S) = 0 = [InN - In A^ + a + (3E^} dA^

(11)

which can be true if and only if 0 = l n A r - l n A ^ + Q ; + /3£M

(12)

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which can be rewritten as N - ^ = e x p (a) exp (/?£„).

(13)

Solving for constants implies E

M

^

N

=

E^exp(Q)exp(/3EM) exp (a) = Y^

N

ex

P (-/3-B/x) • (14)

Thus, N^ _ exp (/?£„) N £ M e x p (-/?£„)

l

>

N

where £ exp(—PE^) is denoted as the partition function and - ^u is the probability. Thus if we are considering the probability of an atom's movement from energy state /it, to an energy state v we have the following: A(fi —>v) _ -£_ _ s„exp(-j3S„) _ exp(/3EM) = exp c [/3 ( £ „ - £ ? „ ) ] A(v—*n) "Z exP(ffgy) exp^) " ^ ^ (16) where for a thermodynamic system /3 = -^ , thus i 4 (/x —•» i/)

— exp

fcT

(17)

The Metropolis algorithm states that the more efficient method to accept movements is to enforce that the larger of the two acceptance probabilities, A, be unity. [24] This results in the following acceptance parameters:

^ ^ M ^ 1 ^ " ^ 0 } . (18) (^ 1 otherwise J Therefore a movement is successful if there is a reduction in the total energy of the system as a result of the movement. However, a movement that raises the total energy of the system is still possible but has a finite probability denned by the previous equation. In addition, the calculation of the energy state is done by implementing the Hamiltonian. The Kawasaki simplification recongizes the fact that the single flip dynamics of the system can make the calculation of the energy states much simpler by considering only those lattice points that have been selected for movement.[24] The energy term defined in the statistical mechanics equation has contributions from three types, surface energy (A-Eg), electromigration forces

R. Atkinson

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:

0.9 7 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

'., ,

. 1 . . . .

0

1 . . . . 1 . . . . 1 . .

1

2

3

4

5

£No. Neighbor

Fig. 1.

Probability of atomic movement, kT = 1 and Eg = 1.

(AEE), and elastic energy (AEa). The surface energy contribution is a result of the binding energy between atoms, where,

Es = {No. of Neighbors)

(EB)

(19)

where Eg is the pair potential energy associated with one atom to another. The Electromigration forces create the energy, AEE, as a result of the atom moving through the electromigration force field. The AEa term is the strain energy that results from the applied forces and displacements to the boundary of the specimen. The inclusion of these terms to the statistical mechanics relation is shown in Equation 20. p = exp

AErotai

kT

p = exp

where AETotai

{ANeighbors)EB

= AES + AEE + AEa

+ AEE + AEa

(20)

A graphical representation of p, for AEE = AEa = 0, is shown in Figure 1. Figure 1 completely describes the probability for a specimen that considers temperature effects only. Notice the integer values along the horizontal axis represent the change in number of neighbors as a result of the movement selected. The inclusion of electric and elastic effects would result in a band of energy around each of the locations that define a change in number of neighbors. This energy band represents the region in which the

Multiscale Modeling of Interconnect

2

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4

ANo. Neighbor Fig. 2.

Probability of atomic movement, fcT = 1 and EB = 1-

1

2

3

ANo. Neighbor Fig. 3.

Probability of atomic movements, for fcT = 1 and Eg = 1 and EB = 2.

Erotai may exist and therefore defines the energy bands in which the atomic movements occur. This type of energy band is displayed in Figure 2. From Figure 1, it is readily seen that the relative size of the environmental energies and change in neighbors, (ANeighbor) (EB), energy term determine the response of the atoms to selected movements. The distance along the x-axis between locations that define motions is a function of the pair potential energy, EB- For example, if the pair potential energy were equal to 2, the probability decreases for the same ANeighbor. A comparison plot is shown in Figure 3.

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The comparison plot shown in Figure 3 displays how the probabilities for varying EB can be altered significantly. The acception or rejection of randomly selected atoms is based on the probability defined by Equation 20 and the generation of a random number. 2.2. Continuum

Formulation

The Hamiltonian defined in Eq. 1 requires the computation of the electric potential density and elastic energy density at each of the occupied lattice sites. These contributions are obtained by setting up subrogate boundary values problems where the boundaries are defined and continuously updated by the occupancy map of the Monte Carlo (MC) calculation. Different continuum techniques can be used to compute these fields, including Finite Differences (FD) and Finite Elements (FE). In this article, we present how both FD and FE can be effectively coupled with MC. We use FD for solving the electric field and FE for solving the elastic one. 2.2.1. Electric field by finite differences The contribution of the electric field to the motion probability of the atoms comes as a result of the energy change associated with an atom that moves through that electric field. This energy change can be thought of as a mechanical work due to the motion of the atom through a vector force field that is directly related to the gradient of the electric field. This net force experienced by an atom due to electron flow is referred to as electromigration force. These electromigration forces are a function of the current density in the material, and the effective valence of the material. Recall, the electromigration forces that have been used in simulations by numerous researches is of the following form and initially presented by J. R. Black in 1969 [6]: FE = -Z*epj

(21)

7 = cVfcjs

(22)

where

and p=-.

(23) c

Therefore FB =

-Z*e\7$E

(24)

Multiscale Modeling of Interconnect

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343

where V $ s is obtained from the solution of the Laplace equation. V2$E

= 0

(25)

The determination of the electromigration forces which drive mass transport is done by solving the Laplace differential equation in conjunction with specified applied boundary conditions. The boundary condition used in this program is applied electric potential. Applied current density is also a possible boundary condition that can be used to obtain a solution for this electrical problem. Applied potential at the boundary is the preferred boundary condition since the enforcement of an applied current density at a boundary results in surfaces at or around a boundary having an artificially applied current density which is effectively specifying an electromigration force at that point. Electromigration force is the desired dependent variable and should not be an applied boundary condition. The variation of the applied potential boundary conditions enables the program to simulate any desired current density. The solution of Laplace's equation across the entire lattice space is accomplished by first establishing a grid over the lattice space. Laplace's equation is a continuum solution to the electrical problem which is why the lattice space must be converted to a grid that can be operated on in a continuum fashion. These grid points house the electric potential values. Also, each grid is identical to one another. Consider an FCC crystal with its (111) plane normal to the line interconnect path and its [10T] direction parallel to the line interconnect path. This crystal has a grid unit defined according to the underlaying lattice space, see Figure 4. This regular and repeatable grid system is oriented as follows: From Figure 4 it is clear that each grid unit has a total of 4 atoms inside its boundaries. There are two atoms completely inside (4 and 5), two that have half of their area inside (2 and 7), and four that have one quarter of their area inside the boundary (1, 3, 6, and 8). This totals four atoms per grid unit. A grid is considered to be populated with atoms if there exists enough atoms, or portions of atoms, inside the grid boundary to total .75 of an atom. In addition, if the grid has any neighboring grid units that are considered not populated, the numerical solution of the Laplace equation is altered to accommodate this fact. Consider a grouping of grids as shown in Figure 4. The numerical representation of the Laplace equation at a grid point that has all of its neighboring grid points populated is given as follows.

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Cuitino

dx

Fig. 4.

Electric Field grid overlayed on Lattice Space.

o4 o,

o2

$ 0

3>3

Fig. 5.

Electric Field grid with assigned Electric Potentials.

<92$ 9a;

V $ (at grid point 0) = -^~o 2 +

d2$ ch/2

$1 + $ 2 - 2 $ 0

dx2

$ 3 + $4 - 2 $ 0

+ •

dy2 (26)

This reduces to the following expression solving for $o :

_L_

1 f_\_ 0

"~ 2 Vdx2

+

dy2

-1

$1 + $2 dx2

$3 + $4 dy2

(27)

where Figure 5 details the labeling convention of the electric potentials for each of the 5 grids. If a grid point has a neighboring grid point that is not populated, the governing numerical representation of the Laplace equation

Multiscale Modeling of Interconnect

3>.

<&2

*0

*,

o„

o2

Oo

(a) Fig. 6.

345

O4

* 4

4>.

Lines

(b)

Electric Field grid for insulated boundary cases.

must take into consideration the fact that there now exists an insulated boundary condition along one of the sides of grid 0. Consider grid 1 has fewer than 0.75 atoms inside of it. This grid is then considered unpopulated and its edges considered to be insulated boundaries, as shown in Figure 6 (a). The requirement for an insulated boundary is that the following: j-n =0

(28)

where n is the insulator's surface normal. Notice that Figure 6 (a) has $1 = $o- This satisfies the insulator boundary condition at its surface normal. This insulated boundary condition changes the solution for $0 to the following form:

+

-(£ £)"'(£ ^ J -

(29

»

Similarly for Figure 6 (b),

The complete solution to the electric field is obtained by implementing a Newton-Raphson iterative method. The selection of this approach to determine the solution to the electric field is a function of simplicity and robustness. This method, as opposed to a boundary element method, for solution of the electric field does not get increasingly complex for an increasingly multiply connected specimen. This is of importance when implementing a discrete atomic level simulation since the specimen may have voids being created, breaking up, or coalescing. In addition, the use of a finite difference, or finite element method, is preferred since the true location of interest of the current densities is near the surfaces of the specimen. The

346

R. Atkinson & A. Cuitino

boundary element method's numerical accuracy decreases as the distance from the boundary element to the point of interest becomes comparable with the length of the elements used. Near the surface is precisely where the electromigration occurs due to the electromigration forces, therefore a finite difference approach was employed. After the electric field has been solved for, the task is then to convert this electric field to electromigration forces by the relation defined in Equation 24. Recall that the grids encompass a total of 4 atoms per unit as shown in Figure 4. The forces are assigned to the atoms in such a manner as to have contribution from any grid that an atom may coexist in. Figure 7 shows a set of grid units numbered 0 to 8 with the atoms in grid 0 numbered 1 to 8. The forces are assigned in the following manner: Fatom2

= \{F0

+ F\+£_4

+ F8)

Fatom 2 = 2 \F° + -^4) Fatom 3 = \ (F0 + F~2__+ F~4 + F7) Fatom 4 = FQ -F atom 5 = F 0

Fatoml

= \(F0

(31)

+ F\ + F 3 + F5)

Fatom 7 = 2 (^0 + ^ 3 ) Fatom 8 = 4 [FQ + F2 + F3 + F6)

Figure 8 shows an example of the electric field forces and how they assign to the atoms. The bold vectors denote the force assigned to each grid point, the thin vectors denote the force assigned to each atom, and the circles denote the position of the atoms in the lattice space. The statistical mechanics portion of the program requires the use of energy as the input variable that determines the probability of a successful move. What has been described here, thus far, has dealt with a conversion of electric potential energy to force. So why not use the electric potential directly? This is done for two reasons. Firstly, the electromigration of atoms in metals is a result of the forces due to the electrons bombarding atoms in the material. This resulting force has been quantified by the relation given in Equation 24. This defines the electromigration forces as a function of the electric potential derivative and a material property referred to as effective valence (Z*). Therefore, to consider only the potential would omit certain material-specific effects. The electromigration forces can be thought of in a mechanical sense, in that the energy state of the atom is not changed unless the atom moves. This simply means that the energy associated with

Multiscale Modeling of Interconnect

F4 Y^ 2 Y

F8 F,

A

7

A

3

]

F2 8 )

F3

F5

347

F7

4 W 5 ( 6

Lines

F6

Fig. 7.

Forces at grid points overlayed on atom unit cell.

Fig. 8.

Forces at grid points and lattice points, example.

this move is as follows: AEE

= FE • Ad

(32)

where Ad is the change in position vector from where the a t o m was before the move, to where it is after. T h e second reason it is beneficial to enforce a change from potential energy to mechanical work is due to the fact t h a t the energy on the grid space behaves correctly away from the material/no-material interfaces. T h e use of energy directly from the solution of the Laplace Equation can result in material movement in an incorrect fashion, as discovered in the early development of the electrical portion of the simulation program presented here.

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The definition of the mass transport model used in continuum approaches consider the electromigration forces at the surface of the lines will result in a sweeping of material along the direction the electrons are moving at a rate that is linearly proportional to the magnitude of the current density at that point. The use of the current density here relates directly to the force on the atom, which relates directly to the energy of a movement in that force field, and then finally to the probability of a movement based on energy states. This final energy state change is where the movement probabilities are determined and, in an aggregate fashion, transforms to rates of mass transport. Since the movement probabilities are a function of the energy change inside of an exponential, an atom that is selected to move against the force field will do so based on a probability given by the statistical mechanics relation 20. The use of the statistical mechanics relation combined with the Metropoli and Kawasaki algorithms results in the movement of an atom selected to move downstream automatically, and the movement of an atom selected to move upstream is accepted or rejected based on the energy drop from that movement. From this simplified argument, it is recognized that the statistical mechanics approach results in increasingly higher aggregate mass transport response to increasingly higher current density magnitudes. This is the same trend that the continuum mass transport models enforce, however the mass transport response of a discrete system of atoms requires many Monte Carlo iterations, and can only be understood in an aggregate sense. This study of individual random movements of atoms in an effort to understand the complex aggregate response of the system is precisely the strength of this Monte Carlo statistical mechanics methodology. The evolution of the system of atoms in response to these environmental effects is what is of interest here.

2.2.2. Elastic field by finite elements Stress gradients in a metal affect the flow of vacancies in the bulk as well as surface migration of atoms. The cause of preexisting stress in the line interconnect is a result of the coefficient of thermal expansion (CTE) mismatch between the silicon dioxide substrate and metal line. The CTEs for Si02 and Cu are .05xl0~ 6 and !6.5xlO~6[K~l], respectively. The copper has nearly 300 times the CTE of the silicon dioxide. The creation of the crystal during the cooling of the metal lines after deposition on the substrate results in residual thermal stresses due to the CTE mismatch between the

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substrate and the metal. These preexisting thermal stresses in the line interconnects have associated with them a mass transport phenomena due to the varying elastic energy field in the material as a result of the induced thermal stress. The resulting stresses in the metal lines are in fact tensile since the metal possesses a higher CTE than the substrate which it is bonding to at just below the metal's melting point. The cooling of the metal then results in a difference in the contraction of the metal as compared to the substrate/barrier. The metal contracts more and as a result experiences a tensile stress after it has cooled down to its operating temperature. Mass transport due to electromigration forces can also cause additional mechanical stresses to develop especially if the line is embedded in a multilayered substrate or simply passivated. The resulting diffusion downstream along the interface will result in a compressive stress at hillock locations and tensile stress at void locations. This project is concerned not with these mechanisms of stress evolution, but the effects of an existing stress state on a specimen, such that varying stress states, regardless of cause, can be investigated. At an atomic level, the term stress has no meaning. The atoms possess a pair potential energy as a function of the distance between the atoms. The derivative of this pair potential energy is the force between the two atoms. The elastic energy is stored in the lattice of the atoms by virtue of the distance between atoms. This requires the solution of the equilibrium positions of each atom individually. This is a computationally intensive calculation since after each Monte Carlo iteration a new equilibrium position must be solved for. Elastic effects can be included, with less computational effort, if an overlaid continuum problem is solved as was done with the electric problem. This enables the simulation to include elastic effects without the need of calculating the energy state after every Monte Carlo iteration. The use of a continuum approach to solve for the elastic energy requires the partitioning of the lattice space into elements as done in the electrical problem. In fact, the same grid pattern used to define the location of electric potentials define the elements that are used in the finite element analysis of the specimen. The corners of the grids previously defined in Figure 4 are nodes of the finite elements. The determination of whether an element has material or not is done in the same manner as defined for the electrical problem. If an element has less than 0.75 atoms inside its boundaries, then the element is considered to be empty and have no contribution to the elastic energy of the specimen.

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The constitutive relation used in this approach is that of an isotropic material. The underlaying structure of the lattice clearly would result in a more complex constitutive relation, and not an isotropic constitutive relation. The consideration of the material as isotropic in an effort to reduce computational time is justified since the scope of this research is to establish effectiveness of the statistical mechanics Monte Carlo method. We use here the simplest case to verify the inclusion of elastic energy affects the response of the atoms. Increased complexity of the solution to the electrical and elastic problem is reserved for future efforts. The objective here is to include electric and elastic effects and compare results to experimental and other simulation results. 3. Applications 3.1. Void Migration Electromigration

and Coalescence Forces

Driven

by

This section we study the translation response of a void subjected to different electric fields and temperatures. We limit our analysis to a void of constant initial size of d = 50 ADs. Voids of different sizes are analyzed in [1]. The goal here is to uncover some of the details that lead to the spawning of a leg from an initially circular void and to identify what occurs after the leg has been spawned and is allowed to continue to grow. The simulation test conditions range from temperatures of 600K, 700K, and 800-ftT; and electric current such that ^ ^ has values of 0.08, 0.161, 0.242. There are three simulations with ^ - = 0.08, ~^ = 0.161, ^rf = 0.242 each having a temperature equal to 6 0 0 ^ , 7QQK, and 800K. This results in a two dimensional array of nine simulation results enabling a comparison of temperature effects with constant electric field, and vice versa. The geometric boundary conditions for all nine of these simulations have periodicity boundaries on all sides of the lattice space so any atom may move across any lattice surface. The applied electric field is defined with the cathode on the right vertical edge of the lattice space. 3.1.1. Case 1: Low intensity Figures 9, 10, and 11 display simulation results for T = 600-ftT, 700K, and 800.£sf respectively; and all have an electric field such that ^P 5 - = 0.08. These simulations show that for T = 700-K" and 8001^ the void motion is stable, but for T = 600K the void does not translate in a stable fashion.

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Void motion in an electric field, T = 600/C, = ^ & = 0.08.

For each of these three simulations presented thus far, the only difference in environment has been temperature. Therefore, it is clear that there exists a relationship between temperature, electric field, and void size such that a critical combination exists where a transition from stable to unstable void translation will occur. In addition, notice the void motion is stable for both T = 700K and 800.ftT. This is due to the fact that the increase in temperature only serves to increase the surface mobility of the void. Therefore, it is reasonable to assume that if simulations are run for T > 800K and AJ?B = 0.08 the void would continue to translate in a stable

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Void motion in an electric field, T = 700K,

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fashion. Similarly, if simulations are run for T < 600K and ^ = 0.08 the void motion would continue to display unstable translation behavior.

3.1.2. Case 2: Intermediate

intensity

Figures 12, 13, and 14 display simulation results for T = 600if, 700/^, and 800.ftT respectively; and all have an electric field such that M i 0.161. EB This is an electric field that is twice the intensity of that in the previous three simulations discussed. The void evolution in these three simulations

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Void motion in an electric field, T = 800K, ^^-

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yield more complex results than can be described by the term unstable. In fact, none of these three simulations translate in a stable fashion. They all exhibit an unstable void translation ranging from simple to complex. The simple unstable translation is shown in Figure 12 (T = 700K and A B J? = 0.161). From this simulation, one can see the void change shape from circular to horseshoe-like. This horseshoe shape can be thought of as the result of the growth of two legs from the initial void, one at the top and one at the bottom surface of the void. These legs spawned from the top and bottom of the void, in Figure 12, are very slender and continue to grow in

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Void motion in an electric field, T = 600A",

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length until the supply of empty space by the initial void is depleted. The result being two separate voids whose shape is crack-like and lay parallel to the electrical current direction. The surface of these two slender voids is fairly smooth for T = QQQK. A very similar result is obtained from the simulation where the temperature is increased to 700X. Figure 13 (T = 700K and ^ - = 0.161) shows the same initial horseshoe-like shape change that occurred for T = 600-FtT, as shown in Figure 12. The first notable difference, with the increase in temperature, is the change in leg shape. The shape of the leg, as shown in

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Void motion in an electric field, T = 700K, ^£-

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Figure 12 (T = 600K and ^ ^ = 0.161), is relatively long, thin, and flat. The shape of the leg, as shown in Figure 13 (T = 7 0 0 ^ and ^ff- = 0.161), is shorter, thicker, and rougher than the leg for the lower temperature. Again, the increased surface mobility due to the temperature increase is the reason the voids are smoother and longer for the lower temperature simulation. The second notable difference between the two simulation results is the shape evolution after the increase to T = 700K. The bottom leg of the horseshoe shape that has developed begins to pinch and emit small voids from it. This also occurs for T = BOOK but not until the 15

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Void motion in an electric field, T = 800K, ^—^ = 0.161.

millionth Monte Carlo step. The release of the smaller void from the leg tip occurred at a much earlier stage in the void evolution for T — 700K, again indicating the surface mobility due to the temperature increase allowed for this shedding of voids to occur at a greater frequency. More complex unstable translation occurs for (T = 800K and ^ 0.161), as shown in Figure 14. This simulation shows that if the temperature and electric field combination is large enough, the increase of the surface mobility enables the two legs of the horseshoe to grow toward the centerline of the void and combine. This results in an island of material floating

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in a void. The void then continues to translate upstream and then releases the island of material back into the bulk. The simulation of (T = 800K and 4l£ 0.161) as shown in Figure 14, though not visible by the snapshots shown in the Figure, also releases many subsequent voids from the small legs that appear just after the formation of the horseshoe shape. 3.1.3. Case 3: High intensity Figures 15, 16, and 17 display simulation results for T = 600iC, 700K, and 0.242. 800.fi' respectively; and all have an electric field such that AEE

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Void motion in an electric field, T = 600-fC,

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Void motion in an electric field, T = 700K, —gr-

This is the largest electric field simulated with these conditions. These three simulation results, for AEE 0.242, show the same basic response for each temperature when compared to the three simulations having ^ P ^ =

0.161. All three simulations show unstable void motion and void breakup, as did the simulations for AJrB = 0.161. This is certainly expected since the intensity of the electric field is increased beyond a value that already led to unstable void motion and breakup. Figure 15 (T = 600K and ^ P ^ = 0.242) shows the typical horseshoe shape evolution development as seen in all but one of the unstable void translations, Figure 9 (T = 600A" and ^P^- = 0.08).

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Void motion in an electric field, T = 800K, ^E-

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Figure 15 (T = 600/C and ^ - = 0.242) shows t h a t the shape of the two voids created are similar to those in Figure 12. T h e primary difference between the two simulation results is t h a t the larger electric current led to a higher frequency of void shedding from the legs. Figure 12 (T = QQQK and E E = 0.161) shows only one void being released from the tip of the leg. Therefore, the increased frequency of the release of voids from the leg tip is due to the increase in electric current. Figure 16 (T = 700K and g e = 0.242) also has an increased frequency of void releases from the leg tips in comparison to its smaller electrical current counterpart in Figure 13

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(T = 70QK and ^P^- = 0.161). Again, the cause of this must certainly be due to the increase in electric current from ^p 2 - = 0.161 to - ^ = 0.242. Figure 16 (T = 700K and ^f- = 0.242) shows that after the horseshoe is well formed, the two legs grow toward the center-line of the void at a steeper angle than seen in any of the other simulations that do not form an island of material inside the void. If the simulation were allowed to continue, the two voids would certainly begin to interact. The steepness of the angle at which the two newly formed voids grow toward the center-line is a function of both the temperature and the electrical current intensity. Figure 17 (T = 800K and ^ = 0.242), like Figure 14 (T = 800K and ^ P ^ = 0.161), evolves into a void with an island of material inside of it. This results when the current density and mobility are such that the material is swept away so fast that rounding is impossible. Like Figure 14 (T = 800K and ^ f = 0.161), the simulation for (T = 8 0 0 ^ and A B g = 0.242) releases subsequent voids from the horseshoe legs. The size of the voids that are shed from the horseshoe legs for (T = 800K and gE = 0.242) are smaller than for any of the simulation combinations of temperature and electrical current. 3.1.4. Void shedding Figure 18 shows each of the nine simulations at the instance they shed their first subsequent void into the bulk. This array of snapshots enables one to see the effects of the combination of temperature and electric current on the size of voids that are released. Consider the middle row of snapshots in Figure 18. This row shows the effects of temperature on void release in a constant electric field. This clearly shows that the size of the void that is released is inversely proportional to the temperature. Similarly, the top row in Figure 18 shows the same relationship between temperature and void size for a constant electric current. Considering the comparison of columns in Figure 18 also leads to the conclusion that there exists a relationship between the size of a void that is released and the electric current. Each of the columns in Figure 18 show a decrease in released void size as the electric current is increased. Therefore, the relationship between released void size and electric current is also inversely proportional. The frequency at which voids are shed is also a function of temperature and current. An increase in either prompts an increase of shedding frequency. In summary, the results from this section lead to the conclusion that combinations of temperature and current densities lead to a transition from

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stable void motion to unstable void motion. Stable void motion is characterized by the defect's ability to translate through the bulk without changing shape to the extent that the void cannot travel as one unit. Unstable void motion is characterized by a change in shape that leads to the formation of legs from the top, bottom, or both top and bottom sides of the void. These legs grow in such a way as to give the shape of the void a horseshoe-like geometry. This horseshoe geometry then results in two non-connected voids

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or an interaction between the two legs resulting in an island of material inside of a void. Whether the legs become two separate voids or combine and form an island of material inside of the void is a function of the temperature and current density. The results observed here are in fact consistent with other computational investigations of the effect of electrical current on preexisting defects in interconnect including the work of [5] [38] [41]. 3.2. Void Migration Electromigration

and Coalescence Driven and Stress Gradients

by

Concurrent

The effects of electromigration are further coupled with the effects of stress driven mass transport. The difference in the CTE between the conducting layer, barrier layer, and substrate results in a constant state of stress after the deposition and annealing is complete. The stress state that exists in the line certainly plays a role in the creation of defects at the interface of the substrate. These initial surface defects provide yet another source and sink for defects in the lines. The fact remains that the continual decrease in line interconnect size has brought the technology to a crossroads. The techniques being used to create these line interconnects is exceeding the capabilities of the materials being used. It is possible to have the line interconnects be fractured and unusable after the creation process is complete and no burn-through testing has even occurred. This signals the importance of the consideration of the thermal stresses that arise during normal use of any electronic device that possesses small features. This section considers the effect of stress driven mass transport based on the strain energy density associated with an atom selected for movement. Recall that the strain energy density for element i is given by Ui = \~d? [ kidVid~.

(33)

This strain energy density is then easily converted to energy based on the number of atoms inside of the element and the area of the element. The approach used in this investigation utilized a regular grid which makes up the elements used in the numerical calculation of the finite elements. Inside each element is four atoms, therefore the strain energy per atom inside of element i is ^

= ~ -

(34)

where Ui is the strain energy density per atom, and A is the area of the element.

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All simulations done in this investigation have applied displacement boundary conditions on each of the vertical edges of the lattice space, and applied force boundary conditions (zero) at both horizontal edges of the lattice space. In this investigation, the stress state is defined in terms of an atom's energy state as it would be in a rectangular line interconnect (under uniaxial applied strain) with no defects on its surface or in its interior. This is defined in the same spirit as it is for electric current, per unit binding energy unit (jjf-)- This dimensionless ratio, as did for electric current ( ^ £ ) , reduces the elastic effects to a form that makes the comparison of effects due to simple surface diffusion and stress driven migration more logical for a statistical mechanics approach such as this. It is important to recognize the fact that even though there exists a nonzero strain energy at each occupied lattice site, the driving force behind the mass transport due to elastic effects is the gradient of the elastic energy field. For example, a large specimen under uniaxial tension has a homogeneous strain energy density and provides no preferential direction for mass transport. However, if there were a defect in that large specimen, then there would be a non-homogeneous strain energy density resulting in a nonzero gradient near the defect. However, the effects of the defect decay as the distance from the defect increases. The result of such a localized effect in a specimen means the stress driven mass transport is a local phenomena based on the presence of a stress gradient. Therefore, in the simplified 2-D plane stress model with uniaxial (x direction) applied displacement boundary conditions the resulting mass transport due to the inclusion of elastic effects is a severely localized one. The result, as seen in the simulation results, is a crack-like defect spawned at the location of greatest strain energy.

3.2.1. Case 1: High stress gradients with no electromigration In this simulation we including stress effects. The lattice space of (x = 250, y = 250) atoms (copper) has periodic boundary conditions on all edges. The temperature of the specimen is 700/^A uniaxial strain is applied in the x direction such that J 2 - = 0.30. A single void of 60 atomic diameters is placed at x = 75. This combination of environmental effects and boundary conditions simulates the effects of a small void in the bulk during annealing. The temperature and ^f- term can be adjusted to simulate variation in operating temperature and CTE mismatch, which is directly related to the stress.

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Void evolution under uniaxial applied stress, such that -^f-

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E

Figure 19 shows the simulation results for the case just described. Initially, the void begins to facet, as expected. Then the locations with the highest strain energy density, the corners of the facets, begin to push away material from those locations. The crystallographic orientation offset simulated (9c = 0°) provides a chance for growth along four corners since the uniaxial applied stress in the x direction results in the maximum strain energy occurring at the top and bottom of the defect. Consider the effect of an offset of 30°, the result of which would be much simpler than the offset

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of 0°. If the crystal offset were set to 30° the initial faceting would result in two corners at the top and bottom of the void, resulting in no ambiguity as to where the maximum strain energy would occur. The material being swept away from the corners of the facets results in the growth of crack-like appendages from those locations, which continue to grow throughout the simulation. There are a few continuum simulations that have included the effects of stress coupled with electromigration. For example the work of Xia et al. (1997) [44] and Bathe et a/.(2000) [5], which are in general agreement with the simulations presented in here. 3.2.2. Case 2: High stress gradients with electromigration The next step is to consider the combination of electromigration and stress driven mass transport simultaneously. The simulation results shown in Figure 20 possess all of the same geometric and environmental initial conditions. The temperature is 700X, a uniaxial strain (x direction) is applied such that •^2- = 0.30, there exists an applied electric potential such that ^£- = 0.08, and the crystal offset is 0°. The initial growth location of the appendages from the voids are the same in both this simulation and the previous one. The appendages, with the inclusion of electric current, begin to sway upwind giving a tentacle-like appearance. The result of this swaying of these crack-like defects would inevitably result in the pinching of the appendage at some point, thus shedding a void. With the inclusion of stress, the top of the void begins to show a distinct tilt that it would not otherwise display with 9c = 0°. The inclusion of elastic effects results in a change in the mass flow rate through the point of highest strain energy density. This effectively reduces the surface migration from the windward side of the void to the leeward side resulting in the void spawning these lengthy crack-like appendages without much translation of the void itself.

3.2.3. Case 3: Low stress gradients with electromigration Figure 21 shows the simulation results for a specimen that is identical to the previous, see Figure 20. The only difference is the applied displacement boundary condition is such that jjf- = 0.15. The evolution of the void in this simulation is much different than the simulation with -ff0- = 0.30. The simulation evolution, for •^2- = 0.15, requires many more Monte Carlo steps to even initiate the crack-like defect. Only a single appendage grows

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Void evolution under uniaxial applied stress, such that -^f- = 0.30, applied

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from the void in this simulation. Recall, the previous simulation resulted in three appendages growing immediately. This single appendage is spawned from the bottom left side of the void and begins to exhibit the same swaying upwind that was seen in Figure 20. The swaying of the appendage in Figure 21 is on a smaller scale and the appendage length is certainly much smaller than those seen in Figure 20. The pinning of mass transport at the tip of the appendage that is growing in Figure 21 begins to enable the bottom surface of the void to grow perpendicular to the electric current, which

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Void evolution under uniaxial applied stress, such that -g2- = 0.15, applied

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is similarly displayed in Figure 20. This growing of the bottom and top surfaces perpendicular to the direction of the electric current, with an offset of 8c = 0 ° , was not possible when elastic effects were not considered. In summary, the predictions from this mixed discrete-continuum approach, as expected, differ from isotropic continuum calculation[44]. The

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main differences are the faceting of the specimen due to its inherent sixfold symmetry of the underlying lattice.

3.3. Void-Surface Forces

Interaction

under

Electromigration

The effects of a void near an external surface are investigated in this section. The void is placed at varying distances away from the initially flat external surface in an effort to see the manner in which the void and the surface will interact. There exists a distance between the void and surface where the interaction between them is strong enough to force the absorption of the void into the external surface. The reason this particular scenario is of interest is since the failure of the line interconnects is not due solely to surface migration. The collection and interaction of defects at fixed locations, such as grain boundaries and vias, is also a potential cause of failure because they serve as a sink for vacancies and voids moving in the bulk. In addition, line interconnects that are used in devices are passivated, whereas the majority of experimental investigations into line interconnect failure have considered unpassivated lines. Consider a defect formed at the interface of the passivation layer and the conducting material. This defect, if in an unpassivated line, could sweep material away from it along the external surface resulting in the size of the defect increasing because the material was able to be relocated. The consideration of the passivation layer restricts this particular scenario to a certain degree because the material is encased and cannot be so easily redistributed along the interface as it can be along the free surface. This passivation layer requires the transportation of vacancies and voids through the bulk in order for the size of a defect along the passivation interface to grow. This results in a 2 to 16 times increase in lifetime for copper line interconnects with a polymide passivation layer, as stated by Jiang and Chiou [25]. Therefore, the interaction between voids and surface defects is of increased concern when considering passivated lines since the surface migration is restricted in this case. The lattice space for these simulations is (x = 250, y = 250) atoms (copper) with periodic boundary conditions along the vertical edge of the lattice space. The horizontal edge of the lattice space is set to no periodicity in order to allow a flat surface for the void to interact with. The void's diameter is 50 ADs and is placed at three different distances from the surface such that the distance between the edge of the void and the surface, a, is

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12, 6, and 3 ADs. The applied electric potential is such that ^ ^ = 0.08 and the temperature is set to 700-ftT. 3.3.1. Case 1: Void far from surface (12 atomic spacing) Figure 22 shows the simulation results for the void that is set 12 ADs away from the surface. The ratio of the radius of the void to the distance a is ^ = 2.083. The void begins to change its geometry by sweeping material away from its bottom surface faster than at its top surface, resulting in the formation of a leg similar to that seen in Figure 9. This is an expected result since the current density in the material between the void and the external surface is larger than the current density at the top surface of the void. The external surface is also affected by the presence of a void so near to it. The material is swept away at the closest point between the void and the external surface resulting in the formation of a ramp along the external surface. The leg grows along side the ramp and eventually the leg is pinched and forms a void which translates in a stable fashion. After the release of this void, another leg begins to grow from the location the first leg pinched. The reason the resulting void is shedding another void is because of the geometry of the pinched region. The pinched region is long enough to foster the growth of yet another leg that gives rise to the release of yet another void from the initial one. Though the void and external surface are clearly affecting each others evolution, the void never combines with the external surface in this simulation. 3.3.2. Case 2: Intermediate distance (6 atomic spacing) Figure 23 shows the simulation results for the same initial conditions described in the previous simulation, except the void is brought closer to the external surface and is a distance of 6 ADs from the external surface, — = 4.167. The void in this simulation also begins its evolution by spawning a leg at its bottom surface. This leg, as in the previous simulation, grows to a point and is then pinched. Thus, it releases a second void into the bulk. However, there are some notable differences between the simulation results in Figure 22 and 23. First, the size of the void that was shed decreased as the void was brought closer to the external surface. This is due to the increase in current density between the void and the external surface. This increase in current density is due to the fact that the void is now closer to the external surface, providing less material in which the current can flow. These effects

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Fig. 22. Void motion near a surface in an electric field, T = 70QK, 5 = 2.083.

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Second, after the release of the void, the pinched region still attached to the initial void seemed to be reabsorbed back into the void. The simulation shown in Figure 22 resulted in a second void being released from the pinched location. The fact that the pinched region seemed to be reabsorbed into the void without spawning another leg can be attributed to the random surface shape inherent in a Monte Carlo process. More specifically, the surface just

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after the release of the voids, in both Figures 22 and 23, are different. The bottom surface of the void, just after the release of the first void, is more flat for the simulation in Figure 22. The result being the swift passing of material along that surface from the pinched region. In comparison, the bottom surface of the void in Figure 23 just after the pinching is not smooth and has to go through a redistribution of material before becoming flat.

3.3.3. Case 3: Void near to surface (3 atomic spacing) Figure 24 shows the simulation results for a void that is 3 ADs away from the external surface, ^ = 8.333. The void and the external surface interact almost immediately in this simulation. The initial distance that resulted in combination is so small that it can be argued that the inherent randomness of the external surface and void could have led to combination without the inclusion of the electric current.

3.3.4. The role of lattice orientation Schimschak and Krug (2000) [38] performed exactly this simulation using ^ = 2 and ~ = 3. The differences between the simulations can be explained based on the different assumptions for in the surface diffusivity. The present simulations consider anisotropic lattice diffusion while in [38] isotropy is assumed. Recall, the use of a hexagonal lattice results in a surface diffusivity that has a six-fold anisotropy. Therefore, the orientation used in this investigation is such that it tends to elongate the initial voids and spawn legs that grow parallel to the line interconnect. If the crystal is rotated 30°, the interaction between the void and external surface would occur without the need to bring them as close. This is clear from the simulation done with the crystal offset (6 c) by 30° as shown in Figure 25. This simulation shown in Figure 25 has identical initial shape, boundary conditions, and environmental variables used in the simulation shown in Figure 9, T = 600K and ^ = 0.08. The only difference is the 6C = 30°. The resulting evolution of the void geometry after the rotation of the crystal is clearly different. The void, with the 30° offset in the crystal did not become elongated like the void with no offset did. This is due to the new relative orientation between the line interconnect and the crystal. The result being a more triangular shape with the top and bottom of the initially circular void growing at an angle to the line interconnect, as opposed to parallel to it as displayed with no crystal offset.

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Fig. 24. Void motion near a surface in an electric field, T = 70QK, ^ =8.333.

=

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T h e propensity of this crystal orientation to grow defects t h a t tend to expand along the line-width leads to the conclusion t h a t this orientation is more capable of fostering void breakup and surface defect growth across the line, thus resulting in shorter times to failure of the line interconnect. This idea is supported by all of the continuum techniques t h a t simulate electromigration with surface diffusivity anisotropy. [41] [30] [20] [32] [5]

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Fig. 25. Void motion in an electric field with a crystal offset of 30°, T = 600.K, and ^ = 0.08.

[38] [18] [3] [34] This idea is also validated by experimental work by Chu et al.[ll] which clearly showed that the crystalographic orientation with the offset of 30° is significantly less resistant to the causes of failure than when the 9c = 0°. In fact, Chu et al.labeled the 30° offset as the preferred orientation for failure.

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In summary, The three Monte Carlo simulations utilized a 0° offset which aligns the [101] direction parallel to the electric current flow direction. This, as stated by Chu et al., is the most likely orientation to result in late failure of the line interconnect. Conversely, the 9c = 30° puts the [101] direction perpendicular to the electric current flow and is less resistant to the effects that lead to failure of the line interconnect. The Monte Carlo simulation results seen in Figures 9 and 25 show two identical initial conditions, except Figure 9 has it's [lOl] direction parallel (9c = 0°) to the electric current flow direction and Figure 25 has it's [lOl] direction at an angle (9c = 30°) to the electric current flow direction. The comparison of results shows that, for 9c = 0°, there is a distinct tendency for the void to elongate and deform in such a way that it becomes flatter than its initial shape. The simulation results for 9c = 30° shows the void becomes more elongated, but in the direction perpendicular to the electric current flow. The propensity for certain orientations to be more resistant to the growth of a transgranular slit is also evidenced in each of the continuum simulation papers that included surface diffusivity anisotropy. The orientation of the surface diffusivity anisotropy used in the continuum papers has the same conclusion regarding the preferred orientation of a crystal in order to slow the growth of a transgranular slit. A consistent factor in the orientations that led to shorter times to failure, regardless of the degree of the anisotropy, is the orientation which has its surface tangent of its most mobile surface parallel to the electric current flow. This due to the fact that the ability to sweep material away from the more mobile surfaces means the defect will grow in that direction. If that direction is transgranular, as it is for the 9c = 30°, then the surface at the top and bottom of the void will have the largest magnitude of current density acting on it and simultaneously be the surface that can resist the movement of its surface atoms the least. This is precisely why the void elongates in the vertical direction for 9c = 30°, and elongates in the horizontal direction for 9C = 0°.

3.4. Interaction

with Grain

Boundaries

This section investigates the effects of grain boundaries on the mass transport of material without and with electric current present. The simulations of the grain boundary scenarios consist of a vertical grain boundary with no electric current, and a simulation with electric current and a grain boundary with its top edge tilted 20° in the same direction as the electron flow, which

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in our simulations is always right to left. There have been several works, using a continuum representation of the material, done to investigate the effects of a grain boundary on mass transport at the interface. The model of the grain boundary used in these Monte Carlo simulations is simply a thin region of the lattice space where the binding energy of the atoms in that region is less than in the bulk.

3.4.1. Thermal grooving Figure 26 shows the simulation results for a vertical grain boundary in a lattice that is (x = 100, y = 120) atoms (copper). There is a periodic boundary condition along the vertical edges of the lattice space. There is a free surface a the bottom edge of the lattice space. The top surface of the lattice space is not periodic but is adjusted to keep vacancies from collecting at the top edge of the lattice space. The atoms that reside in the thin region (2 ADs wide) deemed grain boundary has an adjusted binding energy of 0.85EB, there is no applied electric current, and the temperature is set to 600^. The simulation results shown in Figure 26 shows the creation of a groove appearing with the point of the groove aligned precisely with the grain boundary region. This is a logical result, and is in agreement with the previous works that have utilized a continuum representation to model this situation. [4] [33] [27] Results from Sun and Suo [4] show a similar grooving effect occurring at the grain boundary edge. The similarity of this type of surface evolution in both methods, continuum and discrete, is a close one. The driving initial condition required by the continuum simulation is the angle of inclination of the surface adjacent to the grain boundary. This angle of inclination is a function of the surface tension of both the grain boundary and the surface neighboring it. The continuum simulations [33] [27] utilize that this angle is a constant and is what drives the surface migration of material. The atomic simulation, in effect, is doing the same thing by denning the changing binding energy of the atoms based on where in the lattice space they reside. The angle of inclination shown in Figure 26 remains fairly constant in nearly all of the snapshots. The inherent randomness of the atomic motion keeps the grain boundary interface geometry changing but most certainly has a tendency to orient itself with a constant angle of inclination. Also, shown in Figure 26, there were instances where the tip of the groove formed vacancies and voids just behind it and these newly created defects then walk along the grain boundary until

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Grain boundary triple point evolution, T = 6Q0K, and Grain Boundary binding 0.85EB-

being released into the bulk. Therefore, the grain boundary is yet another vacancy source in a line interconnect. 3.4.2. Void trapping at and emission from grain boundaries Figure 27 shows the simulation results for a tilted grain boundary (20°) with an applied electric current given by — ^ = 0.08 and a temperature

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Fig. 27. Slanted (20°) grain boundary triple point evolution, T = &00K. Leeward grain boundary binding energy = Eg, windward grain boundary energy = 1.15EB-

of 600K . The lattice space is (x = 450, y — 210) atoms (copper) with the same periodicity boundary conditions as in the previous simulation. Figure 28 shows an identical simulation, the only change being an increase in temperature to 700.ftT. The binding energy of the atoms on the left side of the grain boundary is simply EB- The binding energy of the atoms on the right side of the grain boundary is set to 1.15EB- The grain boundary

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Fig. 28. Slanted (20°) grain boundary triple point evolution, T = 700K. Leeward grain boundary binding energy = EB, windward grain boundary energy = 1.15-Ea-

is tilted in an effort to determine if defects that accumulate at the grain boundary would show a tendency to walk along the interface if the dot product between the grain boundary tangent and the electric current is nonzero. A small void having a diameter of 16 atoms is placed on left side of the grain boundary. The electron flow is from right to left, therefore the void and any vacancies spawned from it will travel to the grain boundary

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and be forced to interact with it. It is from these accumulations at the grain boundary, away from the free surface, that any vertical movement of the voids can be identified. Prom the results of the simulations shown in Figures 27 and 28, the small void on the left side of the grain boundary free surface begins to translate in a stable fashion toward the grain boundary. However, the void is shedding more vacancies from it than it is absorbing and therefore disappears but is recombined at the grain boundary. There is a formation of two voids along the grain boundary, in addition to the wedge shape being formed at the free surface grain boundary triple point. The two voids that accumulated at the grain boundary interface initially begin to translate along the grain boundary toward the wedge at the base of the specimen. This downward translation continues until the size of the void at the grain boundary becomes so large that the void then begins to elongate in the x direction. This elongation in the x direction then leads to the pinching of the top defect near the grain boundary, thus leaving a smaller defect at the grain boundary and releasing a void into the bulk of the windward grain. The bottom void that formed at the grain boundary also increases in size due to vacancy collection, however the entire void manages to cross the grain boundary leaving no defect at the grain boundary as the other void had. This occurrence is most likely because the lower void had gotten close enough to the wedge being formed that the resulting electric current density in the thin region between the void and the wedge increased and swept along all of the defect. This is along the same line of reasoning used in the previous set of simulations to explain the interaction between a free surface and a void just below that surface. As displayed already by the simulations showed in Figures 22, 23, and 24, the relative orientation of the crystal to the line interconnect is such that seems to prevent the interaction of a surface and a void. Consider the crystal is offset by 30°. The propensity, as shown by Figure 25, is for the void to elongate in the vertical direction. The result being the likelihood of a void near a surface or grain boundary triple point to combine with it instead of being swept away from it. The ability of the void to translate through a grain boundary, instead of being combined with another defect or surface, implies that defects present in the line interconnect will pass through a grain with a zero offset more easily than through a grain with an offset of 30°. This ability to pass the defects along without accumulation results in a longer life time of that region of the line interconnect. The wedge shape that forms in both Figures 27 and 28 is the result of the electric current sweeping material to the right side of the grain boundary,

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and away from the left side of the grain boundary. The resulting effect is an accumulation of material on the windward side of the grain boundary, and a depletion of material on the leeward side of the grain boundary. A continuum simulation of this type done by Sun and Suo [4] shows precisely this phenomena of mass accumulation and depletion. In summary, the effects a grain boundary has on mass transport when combined with electromigration is an area of concern when the reliability of line interconnects is critical. The computational and experimental investigations into the effects of a grain boundary show that the grain boundary is certainly an potential source for failure of a line interconnect, as well are interfaces between substrates, barrier layers, passivation layers, vias, and any other location where between materials exists. The statistical mechanics Monte Carlo method is able to simulate the small scale effects of a grain boundary during electromigration and compares well to continuum numerical results as well as experimental results. These simulations serve as a proof of concept for introducing additional effects such as mobile grain boundaries and multiple interfaces (polycrystals). 4. Conclusions A mixed discrete-continuum paradigm for simulating the process of degradation and failure of interconnect line due to electromigration and stress gradients is proposed. The discrete calculations utilizes a Monte Carlo approach while the continuum calculations use the finite differences and finite elements. The bridging device is a mixed Hamiltonian, where short-range interactions are described discretely while long-range interactions are incorporated via continuous fields. This approach allows for following complex geometries and topologies resulting from defect formation, migration and coalescence as well as interaction with free surfaces and grain boundaries. The methodology naturally incorporates anisotropy in the diffusivity due to the crystal lattice. Several applications cases are presented, ranging from thermal grooving to trapping and emission of voids at grain boundaries. When appropriate the estimates of the present approach are contrasted with other theoretical and numerical estimates, and experimental results. Acknowledgments This work was supported by National Science Foundation CMS-96-10536. Alberto Cuitino thanks the Institute of Mathematical Sciences (IMS) at the National University of Singapore (NUS) for providing such a nice opportunity to visit the IMS as well as NUS.

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