US 20060271888 A1 Abstract In order to design on-chip interconnect structures in a flexible way, a CAD approach is advocated in three dimensions, describing high frequency effects such as current redistribution due to the skin-effect or eddy currents and the occurrence of slow-wave modes. The electromagnetic environment is described by a scalar electric potential and a magnetic vector potential. These potentials are not uniquely defined, and in order to obtain a consistent discretization scheme, a gauge-transformation field is introduced. The displacement current is taken into account to describe current redistribution and a small-signal analysis solution scheme is proposed based upon existing techniques for static fields in semiconductors. In addition methods and apparatus for refining the mesh used for numerical analysis is described.
Claims(1) 1. A method of numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the method comprising:
directly solving the field equations modified by addition of a dummy field by numerical analysis, and outputting at least one parameter relating to a physical property of the system. Description This application is a continuation of, and incorporates by reference in its entirety, U.S. application Ser. No. 10/630,439, filed Jul. 29, 2003, which is a continuation of U.S. application Ser. No. 09/888,868, titled “METHOD AND APPARATUS FOR SIMULATING PHYSICAL FIELDS”, filed Jun. 25, 2001, now U.S. Pat. No. 6,665,849. U.S. Pat. No. 6,665,849: (1) claims the benefit of U.S. Provisional Application No. 60/213,764, filed Jun. 23, 2000, and is hereby incorporated by reference in its entirety; and (2) is a continuation-in-part of U.S. application Ser. No. 09/328,882, titled “A METHOD FOR LOCALLY REFINING A MESH”, filed Jun. 9, 1999, now U.S. Pat. No. 6,453,275, which claims the benefit of provisional Application No. 60/088,679, filed Jun. 9, 1998. The present invention relates to a method and apparatus for simulating fields especially electromagnetic fields, particularly useful in the context of analysis of interconnect structures, is presented. Many problems in engineering, physics and chemistry require solving systems of partial differential equations of the type:
Electrical engineering: ρ is the charge density, J is the current density, S is the external charge source (recombination, generation, . . . ) Structural engineering: Computational Fluid Dynamics: ρ^{(i)}=u^{(i) }the components of the fluid velocity field, J^{(i)}=P−μ/3∇·u^{(i) }the pressure tensor, S^{(i)}=μ/ρ∇^{2}u^{(i)}+F^{(i)}/m the external force, ∂ρ/∂t→∂ρ/∂t={right arrow over (∇)}·{right arrow over (u)} the convective derivative The list is not exhaustive and there exist many examples where problems are reformulated in such a form that their appearance is as in equation (1). An important example is the Laplace equation and Poisson equation, in which
There have been presented a number of methods for solving a set of partial differential equations as given above. All numerical methods start from representing the continuous problem by a discrete problem on a finite set of representative nodes in the domain where one is interested in the solution. In other words a mesh is generated in a predetermined domain. The domain can be almost anything ranging from at least a part of or a cross-section of a car to at least a part of or a cross-section of a semiconductor device. For clarification purposes, the discussion is limited here to two-dimensional domains and two-dimensional meshes. This mesh comprises nodes and lines connecting these nodes. As a result, the domain is divided into two-dimensional elements. The shape of the elements depends amongst others on the coordinate system that is chosen. If for example a Cartesian coordinate system is chosen, the two-dimensional elements are e.g. rectangles or triangles. Using such a mesh, the domain can be introduced in a computer aided design environment for optimization purposes. Concerning the mesh, one of the issues is to perform the optimization using the appropriate amount of nodes at the appropriate location. There is a minimum amount of nodes required in order to ensure that the optimization process leads to the right solution at least within predetermined error margins. On the other hand, if the total amount of nodes increases, the complexity increases and the optimization process slows down or even can fail. Because at the start of the optimization process, the (initial) mesh usually thus not comprise the appropriate amount of nodes, additional nodes have to be created or nodes have to be removed. Adding nodes is called mesh refinement whereas removing nodes is called mesh coarsening. Four methods are discussed. As stated above, for clarification and simplification purposes the ‘language’ of two dimensions is used, but all statements have a translation to three or more dimensions. The finite-difference method is the most straightforward method for putting a set of partial differential equations on a mesh. One divides the coordinate axes into a set of intervals and a mesh is constructed by all coordinate points and replaces the partial derivatives by finite differences. The method has the advantage that it is easy to program, due to the regularity of the mesh. The disadvantage is that during mesh refinement many spurious additional nodes are generated in regions where no mesh refinement is needed. The finite-box method, as e.g. in A. F. Franz, G. A. Franz, S. Selberherr, C. Ringhofer and P. Markowich “Finite Boxes—A Generalization of the Finite-Difference Method Suitable for Semiconductor Device Simulation” IEEE Trans. on Elec. Dev. ED-30, 1070 (1983), is an improvement of the finite-difference method, in the sense that not all mesh lines need to terminate at the domain boundary. The mesh lines may end at a side of a mesh line such that the mesh consists of a collection of boxes, i.e. the elements. However, numerical stability requires that at most one mesh line may terminate at the side of a box. Therefore mesh refinement still generates a number of spurious points. The issue of the numerical stability can be traced to the five-point finite difference rule that is furthermore exploited during the refinement. The finite-element method is a very popular method because of its high flexibility to cover domains of arbitrary shapes with triangles. The choice in favor of triangles is motivated by the fact that each triangle has three nodes and with three points one can parameterize an arbitrary linear function of two variables, i.e. over the element the solution is written as:
In three dimensions one needs four points, i.e. the triangle becomes a tetrahedron. The assembling strategy is also element by element. Sometimes for CPU time saving reasons, one performs a geometrical preprocessing such that the assembling is done link-wise, but this does not effect the element-by-element discretization and assembling. The disadvantage is that programming requires a lot of work in order to allow for submission of arbitrary complicated domains. Furthermore, adaptive meshing is possible but obtuse triangles are easily generated and one must include algorithms to repair these deficiencies, since numerical stability and numerical correctness suffers from obtuse triangles. As a consequence, mesh refinement and in particular adaptive meshing, generates in general spurious nodes. The finite-element method is not restricted to triangles in a plane. Rectangles (and cubes in three dimensions) have become popular. However, the trial functions are always selected in such a way that a unique value is obtained on the interface. This restriction makes sense for representing scalar functions ψ(x,y) on a plane. In the box-integration method, each node is associated with an area (volume) being determined by the nodes located at the closest distance from this node or in other words, the closest neighbouring node in each direction. Next, the flux divergence equation is converted into an integral equation and using Gauss theorem, the flux integral of the surface of each volume is set equal to the volume integral at the right hand side of the equation, i.e. equation 1 becomes
The assembling is done node-wise, i.e. for each node the surface integral is decomposed into contributions to neighboring nodes and the volume integral at the right-hand side is approximated by the volume times the nodal value. The spatial discretization of the equation then becomes
The advantages/disadvantages of the method are similar as for the Finite element method because the control volumes and the finite elements are conjugate or dual meshes. Voronoi tessellation with the Delaunay algorithm is often exploited to generate the control volumes. However, forming and refining the mesh is not the only problem facing the skilled person in the solution of field theory problems. For instance, the on-chip interconnect structure in modern ULSI integrated circuits is a highly complex electromagnetic system. The full structure may connect more than one million transistors that are hosted on a silicon substrate, and containing up to seven metallization layers, and including interconnect splittings, curves, widenings, etc. A structure results with a pronounced three-dimensional character. As a consequence, analytic solution methods have only limited applicability and numerical or computer-aided design methods need to be used. The continuous down-scaling of the pitch implies that parasitic effects become a major design concern. Furthermore, interconnect delay will soon become the main bottleneck for increasing the operation frequencies of the fully integrated circuit. These observations justify an in-depth analysis of the interconnect problem based on the basic physical laws underlying the description of these systems. Whereas in the past it sufficed to extract the parasitic behavior from the low-frequency values of the characteristic parameters, such as the resistance (R), capacitance (C) and inductance (L), knowledge about the modifications of these parameters due to fast variations in time of the fields, i.e. at high frequency, becomes mandatory. A generic method that allows one to obtain the frequency dependence of the characteristic parameters for an interconnect (sub-) system has been a requirement for some time. The highest frequency in which is currently of interest is 50 GHz, which corresponds to a shortest wavelength of the order of one centimeter. However, this is only a current limit. For most of the interconnects with sub-micron widths the characteristic width (length) of the structure is therefore much smaller than the wavelengths under consideration. In this regime one normally neglects the full displacement current, but this view must be refined depending upon the materials used [H. K. Dirks, Quasi-Stationary Fields for Microelectronic Applications, Electrical Engineering, 79, 145-155, 1996]. Interconnect lines are typically parallel to the axes of a Cartesian grid Manhattan like geometry. Although this is no longer true for widenings and splittings in the lines and the vertical connections, i.e. cylindrical vias, most of the structure can be regarded in a first order approximation as consisting of straight orthogonal lines or bricks. The skin effect becomes important for the upper metallization levels where the width of the structures is larger than the skin depth for aluminum or copper, especially at the high frequency part of the spectrum. Eddy currents play an important role in the lossy semiconductor silicon substrate. It is desirable to formulate the equations for the interconnect system in a language that is familiar to the interconnect-designer community. In particular, variables such as the Poisson field should have their usual meaning. For time-dependent fields it can be achieved by selecting a specific gauge fixing. In particular, in the Coulomb gauge, the Poisson equation remains unaltered. The natural choice for the description of interconnect systems is the one that uses the electric scalar potential and the magnetic vector potential. Small signal analysis (AC analysis) has been a successful tool for extracting compact model parameters for devices [S. E. Laux, Techniques for Small-Signal Analysis of Semiconductor devices, IEEE trans. on computer-aided design, 4, 472-481, 1985]. Recently good results were obtained in using small-signal analysis [S. Jenei, private communication, 2000] for the extraction of compact model parameters for the Hasagawa system [H. Hasegawa et al. IEEE Trans. on Microwave Theory and Techniques vol. MTT-19, 869, 1971] and similar methods are currently exploited for the design of spiral inductors. Numerical analysis is well known to the skilled person, e.g. “The finite element method”, Zienkiewicz and Taylor, Butterworth-Heinemann, 2000 or “Numerical Analysis”, Burden and Faires, Brooks/Cole, 2001. Conventional finite difference numerical analysis solves three-dimensional field theory problems that contain the magnetic vector potential by superimposing three scalar fields, representing this vector potential, whereby each scalar value is located at a node of a mesh. Finite difference methods convert partial differential equations into algebraic equations for each node based on finite differences between a node of interest and a number of neighbours. These methods introduce three types of errors. Firstly, there is the error caused by solving for a discrete mesh, which is only an approximation to a continuum. The smaller the mesh the higher the accuracy. Secondly, the finite difference methods require an iterative solution, which is terminated after a certain time—this implies a residual error. Thirdly, the superposition of three scalar fields is only an accurate representation of vector fields when the mesh size is so small that moving from one node to the next in one direction is associated with a negligible change in the field values in the other two dimensions. In such a case small changes of dimension in one direction may be considered as if the values of the field in the other two are constant. Where there are strongly varying fields this criterion can only be met where the mesh spacing is very small, i.e. there are a large number of nodes. Computational intensity increases rapidly with the number of nodes. To a certain extent the computational intensity can be reduced by modifying the size of mesh so that a tight mesh is only used where the divergence of the field requires this. However, varying mesh sizes places limitations on the continuity of the solution resulting in unnecessary nodes being created to provide sufficiently gradual changes. Hence, conventionally a large amount of storage space and high-powered computers are required to achieve an accurate result in a reasonable amount of time. It is an aim of the present invention to provide numerically stable methods and apparatus implementing these methods for simulating (i.e. calculating) field problems, e.g. electromagnetic fields. It is a further aim of the present invention to provide numerically stable methods and apparatus implementing these methods for simulating (i.e. calculating) field problems, e.g. electromagnetic fields which requires less storage space and preferably less computational intensity. The present invention provides a consistent solution scheme for solving field problems especially electromagnetic modeling that is based upon existing semiconductor techniques. A key ingredient in the latter ones is the numerical solutions method based on a suitable finite difference method such as the Newton-Raphson technique for solving non-linear systems. This technique requires the inversion of large sparse matrices, and of course numerical stability demands that the inverse matrices exist. In particular, the finite difference matrix, e.g. a Newton-Raphson matrix should be square and non-singular. The present invention provides a generic method for solving field problems, e.g. simulating electromagnetic fields, and is designed for numerical stability, in particular the solution of partial differential equations by numerical methods. It is an aspect of the invention that it is recognized that in order to obtain a consistent discretization scheme, meaning leading to numerical fit calculations, a dummy transformation field, also denoted gauge transformation field or auxiliary gauge field can be introduced as a dummy field and can ease computation. The dummy field can be introduced due to the non-uniqueness of the electric and magnetic potentials describing the underlying physical phenomena. It is an aspect of the invention that it is recognized that in order to obtain a consistent discretization scheme special caution is taken in the translation of the continuous field equations onto the discrete lattice, comprising of nodes and links. With the generic method high-frequency parasitic effects and the frequency dependence of the characteristic parameters for an interconnect (sub-) system can be studied but the method is not limited thereto. The present invention provides a method for numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the method comprising:
The method can be formalized as follows: a method for simulating fields in or about a device, said method comprising the steps of:
The output of the method is a field related parameter of the device, e.g. an electromagnetic parameter of the device such as a field strength, a resistivity, an inductance, a magnetic field strength, an electric field strength, an energy value. The field equations of the above method may be the Maxwell equations. The dummy field is preferably a scalar field. The present invention also provides a method for numerical analysis of a simulation of a physical system, the physical system being describable by Maxwell's field equations of which the following is a representation:
Where J and ρ are generic functions of the fields, i.e.
the method comprising:
The physical property may be any of the following non-limiting list: an electric current, a current density, a voltage difference, an electric field value, a plot of an electric field, magnetic field value, a plot of a magnetic field, a resistance or a resistivity conductance or a conductivity, a susceptance or a suceptibility, an inductance, an admittance, a capacitance, a charge, a charge density, an energy of an electric or magnetic field, a permittivity, a heat energy, a noise level induced in any part of a device caused by electromagnetic fields, a frequency. The above methods also include a step refining a mesh used in the numerical analysis in accordance with an embodiment of the present invention. The present invention may provide an apparatus for numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the apparatus comprising: means for solving by numerical analysis a modification of the field equations, the modification being an addition of a dummy field, and means for outputting at least one parameter relating to a physical property of the system. The present invention may also provide an apparatus for numerical analysis of a simulation of a physical system, the physical system being describable by Maxwell's field equations of which the following is a representation:
the apparatus comprising:
The present invention may include a data structure for use in numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the field equations being modified by addition of a dummy field, wherein the data structure comprises the simulation of the physical system as a representation of an n-dimensional mesh in a predetermined domain of the physical system, the mesh comprising nodes and links connecting these nodes thereby dividing said domain in n-dimensional first elements whereby each element is defined by 2^{n }nodes, the data structure being stored in a memory and comprising representations of the nodes and the links between nodes, the data structure also including definitions of a parameter of the dummy field associated with the nodes of the mesh. A data structure for use in numerical analysis of a simulation of a physical system, the physical system being describable by Maxwell's field equations of which the following is a representation:
The present invention also includes a program storage device readable by a machine and encoding a program of instructions for executing any of the methods of the present invention. The present invention also includes a computer program product for numerical analysis of a simulation of a physical system, the physical system being describable by field equations in which a parameter is identifiable as a one-form and solving for a field equation corresponding to the parameter results in a singular differential operation, the computer program product comprising: code for solving the field equations modified by addition of a dummy field by numerical analysis, and code for outputting at least one parameter relating to a physical property of the system. The present invention also includes a computer program product for numerical analysis of a simulation of a physical system, the physical system being describable by Maxwell's field equations of which the following is a representation:
the computer program product comprising:
The present invention also includes a method for numerical analysis of a simulation of a physical system, comprising: transmitting from a near location a description of the physical system to a remote location where a processing engine carries out any of the method in accordance with the present invention, and receiving at a near location at least one physical parameter related to the physical system. The modified field equations are:
In the present invention the introduction of the dummy field represented by χ preferably does not modify the vector potential A found from the solution of the modified field equations when compared with the vector potential found from solution of the unmodified field equations. The accuracy of the method may be checked by comparing known algebraic solutions of simple fields with the solution of the method according to the present invention. In the method the step of directly solving the set of modified field equations is performed by discretizing the set of modified field equations onto a mesh with nodes and links between said nodes. For example, the mesh can be a Cartesian mesh. In particular, in the method the vector potential is defined on the links of the mesh. The advantage of associating a field vector with the links and not with the nodes results from the fact that links define a direction given inherently by the form of the mesh. Hence, a field vector field is associated with an atomic vector element of the mesh. This is a more accurate simulation than using the superposition of scalar fields to simulate a field vector. the present invention makes advantageous use of vector elements in the mesh to solve the field equations more accurately. This reduces the number of nodes required to achieve a certain accuracy. This also means that the amount of memory required is reduced as well as speeding up the calculation time. In the method the dummy field is also defined on the nodes of the mesh as it is a scalar field. That is in the finite difference method the nodes are used as the reference points for values of the dummy field. In the method other terms in the modified field equations are expressed in terms of the vector potential and the dummy field. In the method the curl-curl operation on a vector potential on a link is expressed in function of the vector potential on the link and the vector potentials on neighboring links of this link. The curl-curl operation on a vector potential on a link is expressed in function of the vector potential on the link and the vector potentials on links, defined by wings with said link. In the method the step of directly solving can exploit a Newton-Raphson procedure for solving nonlinear equations. In this case it is preferred to select the dummy field in order to have square non-singular matrices in the Newton-Raphson procedure. In the method the boundary conditions may be determined by solving a Maxwell equation in a space with 1 dimension less than the space in which the original field equations are solved. In a further aspect of the invention a method, i.e. the so-called Cube-Assembling Method (CAM), is disclosed for locally refining a n-dimensional mesh in a predetermined domain, wherein the mesh comprises nodes and n−1 planes connecting these nodes thereby dividing said domain in n-dimensional first elements. This method may be advantageously combined with other embodiments of the invention for the solution of field theory equations. The domain can be almost anything ranging from at least a part of a car to at least a part of a semiconductor device. For clarification purposes, the present invention will be described with reference to two-dimensional domains and two-dimensional meshes but the present invention is not limited thereto. The shape of the elements depends amongst others on the coordinate system, which is chosen. By applying a mesh on a domain, the domain can be introduced in a computer aided design environment for optimization purposes. Concerning the mesh, one of the issues is to perform the optimization using the appropriate amount of nodes at the appropriate location. There is a minimum amount of nodes required in order to ensure that the optimization process leads to the right solution at least within predetermined error margins. On the other hand, if the total amount of nodes increases, the complexity increases and the optimization process slows down or even can fail. Because at the start of the optimization process, the (initial) mesh usually thus not comprise the appropriate amount of nodes, additional nodes have to be created or nodes have to be removed during the optimization process. Adding nodes is called mesh refinement whereas removing nodes is called mesh coarsening. The method of the present invention succeeds in adding or removing nodes locally. The assembling is done over the elements, being e.g. squares or cubes or hypercubes dependent of the dimension of the mesh. Like the finite-box method, the CAM method is easy to program, even in higher dimensions. However, the CAM method does not suffer from the restriction that only one line may terminate at the side of a box. According to this aspect of the invention, a method is disclosed for locally refining a n-dimensional mesh in a predetermined domain, wherein the mesh comprises nodes and n−1 planes connecting these nodes thereby dividing said domain in n-dimensional first elements whereby each element is defined by 2^{n }nodes, said method comprising at least the steps of:
In an embodiment of the invention after the mesh is locally refined, this mesh is locally coarsened. In another embodiment of the invention, the refinement is based on an adaptive meshing strategy. In another aspect of the invention, a program storage device is disclosed storing instructions that when executed by a computer perform the method for locally refining a n-dimensional mesh in a predetermined domain, wherein the mesh comprises nodes and n−1 planes connecting these nodes thereby dividing said domain in n-dimensional first elements whereby each element is defined by 2^{n }nodes, said method comprising at least the steps of:
In an aspect of the invention a method is disclosed for optimizing of a predetermined property of a n-dimensional structure, said method comprising the steps of:
In an embodiment of the invention said structure improvements are based on extracting said property from structure characteristics, determined at a subset of said nodes of said mesh. In a further embodiment of the invention said structure characteristics are determined by solving the partial differential equations, describing the physical behavior of said structure, on said mesh. The present invention will now be described with reference to the following drawings. The present invention will be described with reference to certain embodiments and drawings but the present invention is not limited thereto but only by the claims. In particular, the present invention will be described with reference to the solution of electromagnetic field problems especially those associated with semiconductor devices but the skilled person will appreciate that the present invention has application to the solution of field theory problems in general and to the solution of partial differential equations in general. Without being limited by theory, embodiments of the present invention relating to the solution of electromagnetic field equations are based on the following observations. The Maxwell equations formulated in terms of E and B allow for a geometrical interpretation analogous to fluid dynamics. In this picture the electric field E is a one-form, in other words a numerical value is assigned to each path in space. The numerical value corresponds to the work done by the electric field when a charge would move along the path. The magnetic field B is a two-form i.e. a numerical value is assigned to each area element that counts the number B-field flux lines (flows) that pass through the area element. There is a different and more abstract geometrical interpretation of electrodynamics. The fields E and B can be expressed as derivatives of a scalar V and vector potential A:
Now the fields E and B can be viewed as the curvature of a space. This is the space of phases that may be assigned to quantum fields. This curvature interpretation is lacking in the older geometrical picture of electrodynamics. In order to detect the strength of the electromagnetic field it suffices to go around an infinitesimal loop and measure the mismatch between the starting value of the phase factor and the end value of the phase factor. In analytic calculations this interpretation has no serious consequences because these calculations are based comparing infinitesimal changes in the variables going from one position to another. Therefore, the vector potential can be regarded as a field i.e. its dependence on the space-time variables is only local. However, in a computer calculations are made using a grid or mesh of nodes and links and neighboring positions (grid nodes) are always a finite distance apart. Therefore, the round trip along a closed loop for detecting the electromagnetic field consists of line segments that are also of finite length. The phase factor of each line segment depends on the details of the path and therefore the assignment of the vector potential, in accordance with an aspect of the present invention, is done to these paths. In fact, the exact connection between the phase factor, the path C and the vector potential reads as:
Only in the limit of the mesh size going to zero are there no serious consequences. However, the use of very small mesh sizes increases the memory requirement as well as the time for processing all these nodes in a finite difference scheme. Determining the limit as the mesh size goes to zero is not practical in numerical analysis. The present invention presents new solutions of the Maxwell equations arising from the new geometrical interpretation (and any other field equations having similar singularity problems) using the numerical analysis of finite mesh sizes. The new geometrical interpretation of electrodynamics requires assignment of the vector potential A to the links which are the path segments of the grid in the following way:
In general, if a parameter can be defined as a one-form, i.e. a mapping from a line segment to a number, then this parameter should be represented in the computer code as a variable assigned to the links of the grid. This observation can be widened even more: If a parameter can be identified as a two-form it should be assigned to the area elements (plaquettes) of the grid, and if a parameter can be identified as a three-form it should be assigned to volume elements of the grid. A reference providing technical background information of the above geometrical representations is “The Geometry of Physics”, Theodore Frankel, Cambridge Univ. Press, 1997. However, there remains a serious difficulty. The Maxwell equations formulated in terms of the potentials V and A, exhibit a singular behavior with respect to the inversion of the full differential operators. The fact that the differential operator is singular indicates an underlying symmetry. This symmetry is eliminated in accordance with an aspect of the present invention by symmetry breaking conditions. In the present invention a method based on a dummy or auxiliary field (χ), is described to convert the singular behavior of the differential operator into a regular differential operator without altering the physical content of the system of equations. Summarizing: if a parameter can be identified as a one-form and the corresponding coupling between nearest neighbor nodes of a mesh used for numerical analysis results in a singularity problem, the singularity may be alleviated by the inclusion of an auxiliary parameter without altering the physical realization (solution) of the inversion problem. Computer 10 includes a Central Processing Unit (“CPU”) 15, such as a conventional microprocessor of which a Pentium III processor supplied by Intel Corp. USA is only an example, and a number of other units interconnected via system bus 22. The computer 10 includes at least one memory. Memory may include any of a variety of data storage devices known to the skilled person such as random-access memory (“RAM”), read-only memory (“ROM”), non-volatile read/write memory such as a hard disc as known to the skilled person. For example, computer 10 may further include random-access memory (“RAM”) 24, read-only memory (“ROM”) 26, as well as an optional display adapter 27 for connecting system bus 22 to an optional video display terminal 14, and an optional input/output (I/O) adapter 29 for connecting peripheral devices (e.g., disk and tape drives 23) to system bus 22. Video display terminal 14 can be the visual output of computer 10, which can be any suitable display device such as a CRT-based video display well-known in the art of computer hardware. However, with a portable or notebook-based computer, video display terminal 14 can be replaced with a LCD-based or a gas plasma-based flat-panel display. Computer 10 further includes user interface adapter 19 for connecting a keyboard 16, mouse 18, optional speaker 36, as well as allowing optional physical value inputs from physical value capture devices such as sensors 40 of an external system 20. The sensors 40 may be any suitable sensors for capturing physical parameters of system 20. These sensors may include any sensor for capturing relevant physical values required for solution of the field problems, e.g. temperature, pressure, fluid velocity, electric field, magnetic field, electric current, voltage. Additional or alternative sensors 41 for capturing physical parameters of an additional or alternative physical system 21 may also connected to bus 22 via a communication adapter 39 connecting computer 10 to a data network such as the Internet, an Intranet a Local or Wide Area network (LAN or WAN) or a CAN. This allows transmission of physical values or a representation of the physical system to be simulated over a telecommunications network, e.g. entering a description of a physical system at a near location and transmitting it to a remote location, e.g. via the Internet, where a processor carries out a method in accordance with the present invention and returns a parameter relating to the physical system to a near location. The terms “physical value capture device” or “sensor” includes devices which provide values of parameters of a physical system to be simulated. Similarly, physical value capture devices or sensors may include devices for transmitting details of evolving physical systems. The present invention also includes within its scope that the relevant physical values are input directly into the computer using the keyboard 16 or from storage devices such as 23. A parameter control unit 37 of system 20 and/or 21 may also be connected via a communications adapter 38. Parameter control unit 37 may receive an output value from computer 10 running a computer program for numerical analysis in accordance with the present invention or a value representing or derived from such an output value and may be adapted to alter a parameter of physical system 20 and/or system 21 in response to receipt of the output value from computer 10. For example, the dimension of one element of a semiconductor device may be altered based on the output, a material may be changed, e.g. from aluminium to copper, or a material may be modified, e.g. a different doping level in a semiconductor layer, based on the output. Computer 10 also includes a graphical user interface that resides within machine-readable media to direct the operation of computer 10. Any suitable machine-readable media may retain the graphical user interface, such as a random access memory (RAM) 24, a read-only memory (ROM) 26, a magnetic diskette, magnetic tape, or optical disk (the last three being located in disk and tape drives 23). Any suitable operating system and associated graphical user interface (e.g., Microsoft Windows) may direct CPU 15. In addition, computer 10 includes a control program 51 which resides within computer memory storage 52. Control program 51 contains instructions that when executed on CPU 15 carry out the operations described with respect to any of the methods of the present invention. Those skilled in the art will appreciate that the hardware represented in In the example depicted in In one embodiment, the computer 10 includes certain components that can comprise hardware, software, or a combination thereof. For example, the computer 10 includes a solving component for solving equations and an outputting for outputting data. Maxwell Equations The interconnect modeling directly relies upon the Maxwell equations, that describe the temporal and spatial evolution of the electromagnetic fields in media. Gauss' law
Absence of magnetic monopoles
Maxwell-Faraday
Maxwell-Ampere
The following constitutive equations relate the inductances to the field strengths:
The constitutive equation that relates the current J to the electric field and the current densities, is determined by the medium under consideration. For a conductor the current J is given by Ohm's law.
In a dielectric there are no free charges. As a simplifying approximation, the case will be considered of a dielectric medium whose lossy effects can be neglected. In this case, no current continuity equations need to be solved in the dielectric materials and their dielectric constants may be assumed to be real. Although this is a severe restriction, the dielectric materials that are used in back-end processing of semiconductor devices are sufficiently robust against energy absorption, in order to preserve signal integrity at the frequencies under consideration [A. Von Hippel, Dielectric materials and applications, Artech House, Boston, 1995]. In the semiconducting regions, the current J consists of negatively and positively charged carrier currents obeying the current continuity equations.
In here, the charge and current densities are
In order to implement the equations into software algorithms, an electric scalar potential V and a magnetic vector potential A is introduced in the following way. From equation 14 the magnetic induction B may be written as
As will demonstrated in section on the discretization scheme, the field χ is a key ingredient to set up a consistent discretization scheme. Inserting equation 27 into equation 15 yields:
The basic equations can now be summarized as
It should be stressed that so far no approximations have been made. The regular Coulomb gauge corresponds to χ=0 and is convenient for analytic calculations. In particular, after manipulating the term ∇×∇×A as −∇^{2}A+∇(∇·A), the last term vanishes and equation 34 becomes
As indicated above there are strong arguments to associate the field A to links. First of all, from a gauge-theoretical point of view, the field A is the Lie algebra element that describes the phase factor of a path in real space. A successful discretization of gauge theories assigns the group elements, and therefore the gauge fields to links [K. G. Wilson, Confinement of Quarks, Phys. Rev. D10, 2445, 1974]. Another argument in favor of this association is that the vector potential can be identified in differential geometry with a one-form, i.e. a function on vectors, where in accordance with the present invention the vectors are connecting two adjacent grid nodes [T. Frankel, The Geometry of Physics, Cambridge University Press, 1997]. With these arguments in mind a gauge field variable A_{ij}=A·ê_{ij }is associated to each link where ê_{ij }is a unit vector in the direction of the link between nodes i and j. The time evolution can be described either in real time or in the Fourier domain. In one aspect of the present invention the solution to the field equations will be performed in the Fourier domain. In order to smooth the transition in going from the static to the dynamic description, a calculation scheme is provided that generates the usual characteristic parameters (R,C,L,G) that now become dependent on the operation frequency ω. In the Fourier domain the potential description becomes for the selected gauge:
Analogous to the time-dependent analysis of devices, the interconnect system is treated as a multi-port device with a number of ‘stand-by’ operation conditions at the terminals. In particular, these conditions can be imposed as constant voltage biases or as constant current injections. The stand-by conditions are assumed to be static and therefore, firstly, the static or lowest-order solution is found. The frequency dependent solution is then obtained by superposition of the input signals and the stand-by conditions. Since the magnetic field part plays an essential role in the high-frequency analysis, the magnetic field is preferably included from the start such that the appropriate distribution of electric and magnetic energy is present in the lowest order solution. Starting with the equations 36-38, let ω→0. This static solution (V_{0},A_{0}) will correspond to the stand-by conditions. Starting with the static solution, the different independent variables ξ(=A, V, χ, ρ, n, p) may be rewritten as a static part (with subscript index _{0}) and a non-static part, denoted with a superscript hat ˆ i.e., ξ=ξ_{0}+{circumflex over (ξ)}e^{jωt}. Performing a Taylor series expansion and keeping only the linear terms, the result is a linearized system that can be solved to give the next order solution for the charge and current distributions. Static Approach The electrostatic field, V_{0}, is obtained by solving the Poisson equation
For conducting media the latter equation is supplemented by
In order to extract the RCLG parameters of some interconnect sub-structure, its response to a small harmonic perturbation around a given bias operating point is considered. The bias operation point is a solution of the static set of equations. The equations that determine the amplitudes and phases of the harmonic perturbations are obtained by linear perturbation of the full system. Returning to equations 36-38:
For semiconductors:
Here, the electric field dependence of U has been suppressed but this can easily be taken into account. The equations describe the deviation of the system from the static solution, using the potential fields V and A, the gauge transformation field χ and the densities n en p, as independent variables, as is illustrated in Discretization Scheme Because of the specific geometry of the problem, the set of equations is discretized on a regular Cartesian grid having N nodes in each direction. The total number of nodes in D dimensions is M_{nodes}=N^{D}. To each node may be associated D links along the positive directions, and therefore the grid has roughly D N^{D }links. This is ‘roughly’ because nodes at side walls will have less contributions. In fact, there are 2 D sides with each a number of N^{(D-1) }nodes. Half the fraction of side nodes will not contribute a link in the positive direction. Therefore, the precise number of links in the lattice is M_{links}=D N^{D }(1−1/N). As far as the description of the electromagnetic field is concerned, the counting of unknowns for the full lattice results in M_{links }variables (A_{ij}) for the links, and M_{nodes }variables (V_{i}) for the nodes. Since each link (or node) gives rise to one equation, the naive counting is consistent. However, the gauge condition has not yet implemented. The regular Coulomb gauge ∇·A=0, constrains the link degrees of freedom and therefore not all link fields are independent. There are 3N^{3}(1−1/N) link variables and 3N^{3}(1−1/N)+N^{3 }equations, including the constraints. As a consequence, at first sight it seems that one is confronted with an overdetermined system of equations, since each node provides an extra equation for A. However, the translation of the Maxwell-Ampere equation on the lattice leads to a singular matrix, i.e. not all rows are independent. The rank of the corresponding matrix is 3N^{3}(1−1/N), whereas there are 3N^{3}(1−1/N)+N^{3 }rows and 3N^{3}(1−1/N) columns. Such a situation is highly inconvenient for solving non-linear systems of equations. This arises because the source terms are themselves dependent on the fields. The application of the Newton-Raphson method requires that the matrices in the Newton equation be non-singular and square. In accordance with an aspect of the present invention, the non-singular and square form of the Newton matrix can be recovered by introducing the more general gauge ∇·A+∇^{2}χ=0, where an additional field χ, i.e. one unknown per node, is included. Then the number of unknowns and the number of equations match again. In the continuum limit (N→∞), the field χ and one component of A can be eliminated. However, on a discrete finite lattice the auxiliary field is essential for numerical stability. It may be concluded that the specific gauge only serves as a tool to obtain a consistent discretization scheme. It should be emphasized that the inclusion of the gauge-fixing field χ should not lead to unphysical currents. As a consequence, the χ-field should be a solution of ∇χ=0. To summarize: instead of solving the static problem
The implementation of the gauge condition results in a unique solution and simultaneously arrives at a system containing the same number of equations and variables. Hence a square Newton-Raphson matrix is guaranteed while solving the full set of non-linear equations. Differential Operators in Cartesian Grids The div-operator integrated over a test volume ΔV_{i }surrounding a node i can be discretized as a combination of 6 neighboring links.
The grad-operator for a link ij can be discretized as a combination of 2 neighboring nodes. Integrating over a surface S_{ij }perpendicular to the link ij gives
The grad operator for a link ij integrated along the link ij is given by:
The curl-curl operator can be discretized for a link ij as a combination of 12 neighboring links and the link ij itself. As indicated in The div-grad-operator can be discretized ( The fields (V, A, χ) need to be solved throughout the simulation domain, i.e. for a semiconductor device: conductors, semiconducting regions, dielectric regions. The discretization of these equations by means of the box/surface-integration method gives
In a conductor Ohm's law, J=σE applies, or integrated along a link ij:
For the semiconductor environment the Scharfetter-Gummel scheme can be followed [D. L. Scharfetter, H. K. Gummel, Large scale analysis of a silicon Read diode oscillator, IEEE Trans. Elec. Devices, ED, 16, 64-77, 1969]. In this approach, the diffusion equations:
The simulation domain consists of an interconnect (sub-) system possibly extended with a region of air surrounding it. Therefore, a distinction has to be made between boundary conditions for the simulation domain and boundary conditions for the device. For the latter, it is clear that the electric potential V is defined on the metal terminals provided that voltage boundary conditions are used. The boundary conditions for simulation domain are more subtle. The vector potential A, needs a specific approach. It can easily be seen that just solving equation 40 is not possible. Indeed, the left-hand side of the equation is divergence-less, whereas the right hand side has a non-vanishing divergence on the terminals, where current is entering or leaving the structure. In order to solve this paradox, the analogous situation of a continuity equation is considered. In the latter case, the paradox is lifted by explicitly including the external current into the balance equation. For the curl-curl equation it is necessary to explicitly keep track of the external B-field, i.e. by assigning to every link at the surface of the simulation domain, a variable B_{out}. At edges of the domain this field replaces two missing ‘wings’ of the curl-curl operator, whereas at the other links of the domain surface the B-field stands for one missing ‘wing’ ( Furthermore, the magnetic field must be constructed in such a way that its divergence vanishes. For each plaquette on the simulation boundary it implies that
As indicated in the case study, assembling the different matrices gives:
For χ it is clear from comparing equation 62 with 63 that ∇χ must vanish. This leaves one extra degree of freedom, so that the value of χ can be chosen as equal to 0 in one point. With this choice, the values of χ for the other points are considered as dynamical variables, but will result in χ=0 everywhere. Case Studies In order to be able to construct the differential operator matrices, a choice must be made for the numbering of nodes, edges, surfaces and volumes. A straightforward node numbering is chosen. The numbering starts at the corner of the box with the lowest x, y and z indices, following its neighbor nodes along the x-axis, then jumping back to the lowest x index, incrementing the y-value, and finally when the first plane is numbered, z is incremented. For edges, surfaces and volumes, the number associated with each number is given by the number of the node with the smallest node index. Furthermore, S_{ij }is set to 1, and h_{ij }is set to 1, in the following examples. 2×2×2 cube A simple 8 node cube is shown in 16×16×2 cube In order to check the physical consequences of the way the method deals with boundary conditions, a 16×16×2-node cube is simulated in which a conductor box of one volume element is implemented. At four nodes at one side of the conductor the voltage V=1 is applied, while at the other side the voltage is kept V=0. Hence a current will flow, characterized by the solution of equations 42-44 in the conducting area. This solution of J_{0 }determines J_{out }and B_{out }can be found as a solution of equations 90-91. Next it is possible to calculate the magnetic vector potential in the simulation domain by solving equation 40 ( The skilled person will appreciate certain aspects of the above embodiments of the present invention. A method is provided for the description and the analysis of the electromagnetic behavior of on-chip interconnect structures, by using small-signal analysis. This avoids solving Helmholtz equations and still gives us information on the structure to describe effects like the current redistributions, the impact of high frequencies on the characteristic parameters, slow wave modes etc. A formulation of the Maxwell equations is used that is based on a potential approach. Furthermore, the potential fields are assigned to links. This approach has severe consequences for solving the field equations, which are resolved by the method in accordance with the present invention. In order to solve the Maxwell equations a square and non-singular Newton-Raphson matrix is needed, and such matrices are provided by including an extra dummy potential field χ. This field however will not change the physical solution. In fact, a dedicated gauge fixing procedure has been presented to accommodate for the numerical stability. The magnetic vector potential is calculated by solving a curl-curl operator equation. This task has been carried out in a box-like example of a current carrying wire. The simulated magnetic field shows the behavior of a magnetic field generated by a wire and demonstrates the consistency and correctness of the proposed method. Another interesting result concerns the boundary conditions. The inclusion of the latter is dictated by the conversion of the continuum equations to the discretized equations. Consistency of the boundary conditions demands that a separate Maxwell problem be solved in a domain of dimension D−1=2. One aspect of the present invention is the efficient use of memory space. In accordance with an embodiment of the present invention data structures are created in a memory of a computer system which are closely associated with the numerical analysis methods described above. One possible implementation of such data structures is given below. The data structures are a representation of a mesh having links connecting nodes in a mesh structure. The implementation is based on a mesh formed by cubes but the present invention is not limited to cubes. The implementation makes use of pointers however the present invention is not limited to pointer based systems but may include any method of referring to other memory locations. node (see This structure is used to stock nodes in a list.
All properties of the nodes (x, y, z, . . . ) are stored in this structure. The properties can be: V, the Poisson potential, ρ the charge density, N the dopant concentration, n and p the electron and hole concentration, T the temperature and χ the dummy field. In accordance with an aspect of the present invention the zero-forms and three-forms are associated with the nodes of the mesh. The nodes are internally placed in a linked list, where each node points to the next node and the last node points to NULL. Fields
link ( This structure is used to stock links in a list.
All properties of the links are stored in this structure. In particular values for those elements of the fields such as the vector potential A which are associated with links are stored in this structure. The properties can be: A the magnetic vector potential, J, the current density (carrier density in semiconductor substrates), E the electric field. The links are identified by 2 nodes. In accordance with an aspect of the present invention, the one-forms are associated with the links. The links are internally placed in a linked list, where each link points to the next link and the last link points to NULL. A link can have 2 childLinks. The pointers link1 and link2 point to these children. If these pointers are NULL, the link doesn't have any children. Fields
cube (see This structure is used to stock cubes in a list.
The links are identified by number. This is the number of the cube. Internally, the cubes are organized in several linked lists. All cubes with the same generation are stored in the same linked list. Fields
cubeListPointer This structure is used to point to a cubeList.
Internally, cubes are organised in several linked lists. All cubes with the same generation are stored in the same linked list. Something is required to point to all these lists. This is what a cubeListPointer does, it points to a cubeList and to the next cubeListPointer. So the first cubeListPointer points to the cubeList generation 1, the next to the list with generation 2, . . . and so on. Fields
lastNumbers This structure is used to keep track of the last nodenumber, linknumber and cubenumber.
The last nodenumber, linknumber and cubenumber are stored here, so that nodes/links/cubes can easily be appended and the nodenumber/linknumber/cubenumber can be filled in. Fields
The calculation method for the pointers is given below. In the following a detailed description of practical applications of the methods of the present invention are described. In order to extract the RCLG parameters of an interconnect sub-structure, its response to a small harmonic perturbation around a given bias operating point is considered, which is a solution of the static set of equations. The equations that determine the amplitudes and phases of the harmonic perturbations are obtained as linear perturbations of the full system. Returning to equations (36-38), one obtains
Continuing the discussion on boundary conditions above, the vector potential A, needs a specific approach. It can easily be seen that just solving equation 40 is not possible. Indeed, the left hand side of the equation is ∇^{2}χ, whereas the right hand side has a non-vanishing divergence on the terminals, where current is entering or leaving the structure. However, as was argued above the dummy field χ=0 is part of the solution. In order to solve this paradox the analogous situation of a continuity equation is considered. In the latter case, the paradox is lifted by explicitly including the external current into the balance equation. The external currents, impinging perpendicular on the boundary surface of the simulation domain, carry their own circular magnetic field. Such a magnetic field is described by a component of the vector potential parallel to the impinging current. Therefore the boundary condition for the vector potential will be equal to zero for all links that are in the boundary surface, ∂Ω, of the simulation domain, Ω, whereas links pointing orthogonally inwards from the enclosing surface are part of the set of the unknown variables that should be solve,
The boundary condition for the χ-field will be that χ=0 at the enclosing surface. The Laplace problem on a closed surface with these boundary conditions guarantees that χ=0, everywhere. The boundary conditions for the scalar potential V are a mixture of Dirichlet and Neumann boundary conditions. At the contacts Dirichlet boundary conditions are assumed whereas at the remaining part of the enclosing surface Neumann boundary conditions are assumed. This assumption implies that no perpendicular electric field exists for these parts of ∂Ω. If a contact is placed on a semi-conducting region, it is assumed that this contact is also ohmic. Therefore, the boundary condition for a semiconductor contact is
In general, the structures consist of insulating, semiconducting and metallic regions. As a consequence, there will be four types of interface nodes, i.e. insulator/metal interface nodes insulator/semiconductor interface nodes metal/semiconductor interface nodes insulator/semiconductor/metal ‘triple’ points At the metal/semiconductor interface nodes, the idealized interface Schottky contact condition is implemented, as for a boundary condition for a semiconductor region, by setting φ_{p}=φ_{n}=V_{metal}, where V_{metal }is the value of the Poisson potential at the metal side of the interface. The Poisson potential at the semiconductor side of the interface is V_{semi}=V_{metal}−δV, where δV represents the contact potential between the two materials. Using
At a metal/semiconductor interface node there is one variable (V_{metal}) that needs to be solved. The equation for this variable assigned to the node i is the current-continuity equation,
At metal/insulator interface nodes continuity of the Poisson potential is assumed. For these nodes there is, apart from the variables A and χ, one unknown V_{i}, and the corresponding equation is the current-continuity equation. The Poisson equation determines the interface charge, ρ_{i }and can be obtained by post-processing, once V is determined. At insulator/semiconductor interface nodes there are three unknowns to be determined, V, n and p. These variables are treated in the usual way as is done in device simulation tools, i.e. the Poisson equation is solved self-consistently with the current-continuity equations for n and p, while V is continuous at the insulator/semiconductor interface. At triple point nodes, the Poisson potential is triple-valued. One arrives at different values depending on the material in which one approaches the node. For computational convenience the value of the Poisson potential in the insulator at the midpoint between V_{metal }and V_{semi }is taken, i.e.
The interface conditions for the vector field A and the ghost field χ are straightforward. The choice of the gauge condition, equation 38, is independent of specific material parameters. During assembling of equation 37, the current associated to each link is uniquely determined in an earlier iteration of the Gummel loop and therefore for the vector potential (as well as χ) is single-valued. Scaling Considerations In order to program the equations on a computer, an appropriate scaling must be performed. A generalization of the ‘de Mari’ scaling, as described in A. de Mari, An Accurate Numerical Steady-State One Dimensional Solution of the PN-Junction, Solid-State Electronics, 11, 33-58, 1968 is adopted. Let λ be the scaling parameter for lengths, n_{i }the scaling parameter for doping and carrier concentrations and let the thermal voltage V_{T}=[kT/q] act as a scaling parameter for the Poisson field and the fermi levels. Then from Poisson's equation one obtains that
From the constitutive equations for the carrier currents the scaling factor for the mobility, σ_{μ} is obtained:
There is still the freedom to set one scaling parameter. The scaling parameter for the diffusion constant, σ_{D}=1 [(m^{2})/sec] is fixed. Then the scaling parameter for the time, τ, is given by
Furthermore, from the scaling factor for the diffusivity the scaling factor for the current density is obtained:
The frequency scaling factor is inverse to the time scaling factor, i.e. σ_{ω}=[1/(τ)]. The scaling factor for A and χ follows from the generalized formula for the electric field,
The scaling of the curl-curl equation for A leads to the dimensionless constant, K=[1/(c^{2})] ([(λ)/(τ)])^{2 }where c is the speed of light in vacuum. From table I, K is a rather small number that makes it suitable for using it as a perturbation expansion parameter.
After scaling the curl-curl equation takes the following form
The novelty of the new approach for solving the equations for the scalar and the vector potentials is the association of the vector potential variables to the links or connections of the discretization grid. This requires that not only the grid nodes receive a unique pointer, but also the grid links. In order to become familiar with this new situation, a method for assigning unique pointers to the grid nodes and the grid links in Cartesian grids is presented. For a Cartesian grid with N_{node}=k_{x}×k_{y}×k_{z }nodes, a unique node pointer is obtained by
Given a node pointer L_{node}, the vector n can be reconstructed using the following algorithm
In a similar way a unique pointer, L_{link }can be assigned to each link. Given the node n and a direction i=1, 2, 3, a link pointer can be obtained by the following algorithm, since each link is based in some node n, and points in a given positive direction, i.
Using the INT-function, (n,i) can be extracted from L_{link}, as was done for the nodes. Solver Requirements Since a large number of equations need to be solved simultaneously, i.e. Poisson's equation, the current-continuity equations, the curl-curl equation and the equation for χ, both the static, real and imaginary parts, a Gummel iterative procedure is followed for solving this system. In particular, since the frequency, ω is fixed, the problem is still three-dimensional. Whereas the Poisson's equation and the current-continuity equations can be treated similarly as in device simulation programs such as D. L. Scharfetter, H. K. Gummel, Large Scale Analysis of a Silicon Read Diode Oscillator, IEEE Trans. Elec. Devices, ED, 16, 64-77, 1969 and E. M. Buturla, P. E. Cottrell, B. M. Grossman and K. A. Salsburg, Finite-Element Analysis of Semiconductor Devices: the FIELDAY program, IBM Journal on Research and Development, 25, 218-231, 1981, the equations for A and X require a different handling. Furthermore, the largest system that needs to be solved is also the pair of equations for (A, χ). These equations can not be loaded iteratively into the Gummel flow because the ∇x∇x operator would lead to a singular algebraic system. However, the simultaneous assembling with the equation for χ, results in an algebraic system having a regular matrix that can be inverted. After testing a number of different linear solvers with and without pre-conditioning, it turned out that the most robust method was the symmetric successive-over-relaxation (SSOR) pre-conditioner, taking ω_{SOR}=1.2, combined with the conjugate-gradient squared (CGS) iterative solver. The parameter γ in equation 110 was taken equal to one. The memory requirements for the sparse storage of the Newton-Raphson matrix can be obtained as follows. Suppose there are N_{node}=k_{x}×k_{y}×k_{z }nodes and N_{link}=3×k_{x}×k_{y}×k_{z}−(k_{x}×k_{y}+k_{x}×k_{z}+k_{y}×k_{z}) links. Each (interior) link interacts with 12 neighboring links and 2 nodes in the assembling of the curl-curl equation. This implies that a each link generates 1+12+2=15 non-zero entries in the Newton-Raphson matrix. The scalar equation for the χ-variable in each node interacts with 6 χ's in the neighboring nodes and with 6 link variables sited at the links connecting the node to the nearest-neighbor points. Therefore, each χ field induces 1+6+6=13 non-zero entries in the Newton-Raphson matrix. The total number of non-zero entries can be estimated (ignoring surface subtractions) as N_{non} _{ — } _{zero}=15×N_{link}+13×N_{node}. Table II gives a few numerical examples.
A number of examples are presented demonstrating that the proposed potential formulation in terms of the Poisson field V, the vector field A and the dummy field χ, is a viable method to solve the Maxwell field problem. All subtleties related to that formulation, i.e. the positioning of the vector potential on links, and the introduction of the ghost field χ, have already been described above in constructing the solutions of the static equations. Therefore, a series of examples in the static limit are presented. Metal Plug on Highly-Doped Silicon The first example concerns the electromagnetic behavior of a metal plug on (highly-doped) silicon. This example addresses all subtleties that are related to metal-semiconductor, metal-insulator and semiconductor-insulator interfaces as well as triple lines. The simulation region (10×10×10 μm^{3}) consists of two layers. A layer of the silicon (5 μm) is highly doped at the top, using a square mask of 4×4 μm^{2 }at the center. Above the silicon there is 5 μm oxide with a metal plug of 4×4 μm^{2}. In The energies have been calculated in two different ways and good agreement is observed. This confirms that the new methods underlying the field solver are trustworthy. The χ-field is zero within the numerical accuracy, i.e. χ˜O (10^{−14}).
Crossing Wires The second example concerns two crossing wires. This example addresses the three-dimensional aspects of the solver. The structure is depicted in
Square Coaxial Cable To show that also inductance calculations are adequately addressed, the inductance per unit length (L) is calculated of a square coaxial cable as depicted in with l denoting the length of the cable. As expected, for large values of the ratio r=b/a, the numerical result for the square cable approaches the analytical result for a cylindrical cable, L=[(μ_{0 }ln(b/a))/(2π)].
Spiral Inductor A spiral inductor, as shown in In
The above three-dimensional field solution method has been programmed in software for on-chip passive structures. A TCAD software environment was built such that arbitrary Manhattan-like structures can be loaded, calculated and the results can be visualized. The multi-layer stack of a back-end process results in different material interfaces. The inclusion of the interface conditions in the TCAD software was done such that the electromagnetic response is accurately simulated. The treatment of the domain boundaries was done based on the idea that energy should not enter or leave the simulation domain except for the contact ports. The numerical implementation requires that all variables are scaled and the de Mari scaling was extended to include the vector field A and the ghost field χ. Numbering schemes for the nodes and the links were given and the solver requirements have been specified. In calculating the above examples it is wasteful of nodes to have the same mesh size over the whole of the area/volume of interest. In certain areas/volumes, e.g. where the field intensities change rapidly or in areas/volumes of great importance it is preferred to have a smaller mesh size. However, in other areas it is economical in memory size and computing time to have a wider mesh spacing. In a further embodiment of the invention a method is disclosed for refining a mesh. This method may be combined advantageously with the previous field calculation methods or may be used in the calculation of field problems by other methods, e.g. in fluid dynamics, mechanics etc. This embodiment is therefore not limited in its use to the above filed calculation methods. As an example the application of the method to a rectangular 2-dimensional mesh in a predetermined domain will be described but the present invention is not limited to this number of dimensions. The rectangular mesh comprises nodes and one-dimensional planes, i.e. lines called links, connecting these nodes. As a result, the domain is divided into 2-dimensional rectangular first elements whereby each element is defined by 2^{2 }nodes, i.e. rectangles. The assembling is performed over these rectangles which makes sense despite the fact that the rectangles have four nodes. Indeed, if the solution were modeled as a linear function over a rectangle, the number of conditions for finding the coefficients would result into an overdetermined problem. Therefore, the interpolation is dependent on the purpose for which the interpolation is used. For post-processing one may complete the mesh with a Delaunay tessellation and exploit linear interpolation, however, for solving the system of equations, only the end point values of each link or side enter the equations. A link (see The basic features of a CAM algorithm in accordance an embodiment of the present invention are introduced with an example. Suppose one wants to simulate the electrical potential W(x) in a rectangular grid defining some device, given that the electrical system is described by the Poisson equation with D(x)=εE(x) the electrical field and ρ(x) the electrical charge source and x the place coordinate. The rectangular device is represented by a domain D ( For said simulation the above equation is discretized on a rectangular grid or mesh. Said grid divides the domain D in a set of smaller domains D_{i}. The union of said domains D_{i }(D_{1}−D_{13}) gives D. The equation is now written for each node of the grid. This is further illustrated for the node 9, central in the domain D. Around said node, four rectangles D_{5}, D_{7}, D_{6 }and D_{4 }are recognized. In each of said rectangles areas A_{1}, A_{4}, A_{3 }and A_{2 }(shaded in For each of the nodes of the mesh such an equation is written. In each equation the relation between the electrical charge ρ_{i }and the electrical field I_{ij }with the electrical potential is introduced. As such a set of nonlinear equations (ρ_{i }will depend itself nonlinearly on W_{i}) in the variables W_{i}, being the electrical potential at each node of the mesh, is obtained. Said set of equations is then solved by using an iterative procedure such as a Newton-Raphson scheme. In more general terms I_{ij }is denoted a link-current and ρ_{i}×A is denoted source contribution. The CAM algorithm can be written schematically in the following form:
The function solve_on_rectangles can be written schematically in the following form:
Note that other orderings of the do-loops are also possible. The solve_equations routine can be any nonlinear equation solver. According to this embodiment of the invention, a method is disclosed for locally refining a rectangular 2-dimensional mesh in a predetermined domain, wherein the mesh comprises nodes and lines connecting these nodes thereby dividing said domain in 2-dimensional first elements whereby each element is defined by 4 nodes. Particularly, this method, can locally refine an initial mesh comprising 2-dimensional first elements, i.e. rectangles. This method comprises at least the steps of:
This first additional node is created somewhere inside a rectangle. This can be anywhere inside a rectangle and thus not on a link (side) of the rectangle. Particularly, this first additional node can be created in the center of the rectangle. Regardless of the exact location of the first additional node, this first rectangle is split completely in exactly four new rectangles, i.e. second rectangles. In other words, the sum of the areas of these four second rectangles completely coincides with the area of this first rectangle and therefore this first rectangle can be deleted. This also holds in an analogous way for the second additional node. In another embodiment the method of the present invention is embedded in an adaptive meshing strategy. Adaptive meshing is straightforward for n-dimensional meshes comprising n-dimensional elements whereby each element is defined by 2^{n }nodes. Particularly adaptive meshing can be easily implemented for a two-dimensional mesh comprising rectangles. There are no restrictions on the number of links ending on another link as e.g. in the finite-box method. Therefore, extra algorithms for smoothing the mesh after the adaptive meshing are not required because there is no metastasis of spurious nodes into irrelevant regions; in other words the refinement remains locally. In An important problem in simulation is the use of different physical models on different length scales, or alternatively, using the same physical models at different scales but with modified model parameters. In the latter case, the model parameters follow renormalization group flows, as in K. Wilson “Confinement of Quarks” Phys. Rev. D10, 2445 (1974). Both scenarios can be applied in restricted domains, without blurred transition regions, because the refinement scheme does not generate extra nodes which necessitate subsequent smoothing steps. Concerning the issue of numerical stability of the method of the present invention, it is demonstrated that sets of equations of the type
In three dimensions there exists 12 different classes of orthogonal coordinate systems. The orthogonality allows for applying the subdivision of any cell in 4 subcells by taking the mean of the minimum and maximum of the coordinate variables. The orthogonality is a sufficient reason for the guaranty that one does not have to include smoothing nodes. Another interesting application concerns the transition from one coordinate system to another. When using the method of the present invention, there are no restrictions on the number of links ending on another link. Therefore, the absence of smoothing nodes implies that only transition nodes have to be generated at the pre selected interval region. This is done by detecting the nodes in coordinate system A and in the transition region and by assigning refinement nodes to the coordinate system B. In the same way the nodes of coordinate system B lying in the transition region become new nodes for coordinate system A. This is illustrated in In another embodiment of the present invention, a mesh is created using the locally refinement method of the present invention. Thereafter the mesh can be locally coarsened. When solving systems of partial differential equations of the type
The Cube-Assembling Meshing method is now demonstrated with the simulation of a microelectronic structure. A detailed exposure of the adaptive meshing strategy in a realistic application is presented. The selected structure is an implanted diode with side contact. The purpose of this example is two-fold: (1) to demonstrate that the new method gives acceptable results, (2) to demonstrate that the method has been implemented in a two-dimensional device simulator, which has served as a research tool for predicting highly unconventional devices, such as hetero-junction type vertical transistors, multi layer HEMTS as well as more conventional structures, such as CMOS devices in operation points where effects are present which are very difficult to simulate, such as carrier heating. All these structures have in common that commercial software tools do not provide sufficient reliable data for the parameters which are of interest to the process engineer. This situation is due to the fact that software development progresses in parallel with technology development. Having available a source code for the simulation of the internal dynamics inside devices, an ongoing activity has been to improve the algorithms which are exploited in solving the equations underlying the device dynamics. Although many code improvements deal with a gradual extension towards more accurate models to be implemented, occasionally dramatic jumps in code improvements are realized by implementing pieces of code which involve a new understanding of the mathematical machinery which applied. In the past decade it is observed three instances of major breakthrough events in solving the semi-conductor device equations on the computer. In 1988, a method for simulating abrupt hetero structures, by programming finite jumps in the matching conditions of the nodal values of the scalar functions in the finite-element method. In 1993, a method was constructed for obtaining smooth solutions as well as a robust iterative scheme for finding the solutions of the energy-balance equations by realizing that CGS solvers require a semi-definite Newton-Raphson Jacobian matrix. The third ‘quantum leap’ in improving the algorithm for finding the solutions of the semiconductor device equations was invented in 1998. With decreasing device dimensions, there is an increasing need for the inclusion of quantum effects in the simulation method. The laws of quantum-mechanics are of a substantially different character as the laws of classical physics, e.g. even in first order quantization, wave-like equations need to be incorporated, which are very different from Poisson equation and balance equations in general which are of the diffusive type. This situation leads to reconsideration of the discretization schemes, with having in mind a method which decouples the macroscopic physics from the microscopic physics. The underlying idea is that quantum effects are important in particular regions of the device but that other parts the device could be described by the classical methods. Furthermore the quantum regime should be entered by locally refining the mesh in order to take into account the fact that the quantum effect are dominate on a small distance scale. A renormalization group transformation should realize the connection between the macroscopic and microscopic domains. Unfortunately, local mesh refinements in adaptive meshing schemes are always accompanied by spurious nodes which take care of a smooth transition between the fine and the coarse mesh. Such spurious nodes are lethal to the developed ideas for incorporating the quantum physics in the simulation method. Therefore, the question, whether it would be possible to incorporate a discretization scheme for the classical device equations, which may be submitted to local refinement without generating a metastasis of the spurious refinement nodes, has raised and been solved by the development of the CAM-scheme. The method combines the finite-element method with the box-integration method and is applicable in an arbitrary number of dimensions. The method is limited to meshes consisting of the union of patches of orthogonal coordinate frames. The device under consideration consists of a np-diode, whose third dimension is much larger as the first and the second dimension. Therefore, a two-dimensional cross section suffices for the determination of the diode characteristics. In a perpendicular plane, the diode has dimensions 10 times 5 μm squared. A p-type doped region of size 5 times 2.5 μm squared is allocated in the upper left corner with a contact named top, and the remaining region is n-typed doped. There is a side-wall contact named side connected to the n-type region. When the diode is forward biased, a current flow is expected from the side-wall contact to the top contact. This current is not expected to be one-dimensional due to the perpendicular orientation of the contacts. The adaptive meshing algorithm is expected to allocate refinement nodes such that the current paths can be accurately traced. In The input file diode.dat for the simulator PRISM is presented below.
At the first run of a new structure the output of the PRISM structure generator is loaded into the PRISM simulation. This is the file diode.str as indicated in the the second line. The file diode.str is printed below. In the section STRUCTURE-LINES for each line the fourth parameter gives the number of line divisions. With this parameter a crude initial mesh for performing initial calculations towards the final solution are generated. These number can be provided interactively with the structure generator.
The execution of PRISM with the input file diode.dat generates two files related to the mesh construction. The first file diode.sqr is a detailed description which contains all structure information for applying the cube assemble method. Part of the file diode.sqr is printed below.
Furthermore, there is printed a file diode.gpt, which is suitable for drawing the initial mesh using xgnuplot. The plotting is presented in
A summary file of the results is also produced which provides the biases at the contact and the resulting currents. The file is printed below.
Having obtained an initial mesh as well as a zero bias solution, one may proceed along different tracks. First a mesh refinement according to doping criteria and/or electric field criteria may be done. Other criteria do not make much sense at zero bias. One may also first ramp up the bias to some particular value and apply mesh adaption at the final stage. Which method is selected is not relevant for demonstrating the feasibility of the Cube-Assembling Method, since in both cases the method should final mesh should result into a L stable and convergent solution scheme. Which new nodes are ultimately participating depends on the history of the application of the refinement criteria and the order of applying first adaption and then refinement or first ramping and then adaption may effect the presence of final nodes. Following the first option, i.e. applying adaption on the zero-bias solution, refinement according to the doping and electric-field profile is obtained. In
A current-voltage plot is presented in While the invention has been shown and described with reference to preferred embodiments, it will be understood by those skilled in the art that various changes or modifications in form and detail may be made without departing from the scope and spirit of this invention. For example, although the embodiments of the present invention have been described generally with reference to Cartesian grids, the present invention may be applied to any form of grid used in numerical analysis. Further, although the present invention has been described with reference to the numerical analysis of Maxwell's equations it applies equally well to the solution of partial differential equations, including the refinement of the mesh used in such solutions. Referenced by
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