Geometrical SIMULATION AND VIZUALIZATION
IN NANOELECTRONICS PROBLEMS
S.Polyakov, M.Iakobovski
Institute for Mathematical Modeling, Russian Academy of Science, Moscow, Russia
This work has been supported by Russian Foundation for Basic Research (projects № 08-07-00458a, 09-01-00448a).
Contents
2. Geometrical methods and visualization in nanoelectronics problems
3. Simulation of electron transport in quantum structures
4. Simulation of electron transport in vacuum micro- and nanostructures
5. Simulation of nucleation and migration of voids in interconnects of electrical circuits
Application of methods of geometrical modeling and visualization to the modern problems in nanoelectronics is considered. The problem analysis by the example of several specific targets in the area of modeling of non-linear electronic transport in micro- and nanostructures is carried out. The work clearly shows that geometrical modeling and visualization are an integral and essential part of modern computing experiment.
Keywords: Numerical and geometrical simulations, visualization, parallel algorithms, iterative processes, nanoelectronics, electron transport in nanostructures, quantum wires, vacuum micro- and nanoelectronics, field emitter array, semiconductor micro-cathode, charge carrier heating, quasi-hydrodynamic approach.
Development of new quantum devices with the size of active elements of an order of 10 nanometers and less is a perspective direction of modern nanoelectronics. Such devices will be applied in the near future to the creation of systems for processing, storing and displaying data and information, radiation sources with super-high frequency, electronic probing equipment [1.1, 1.2]. Recently, application of quantum devices for creation of supercompact current sources, nanotransistors and computer memory elements is seriously considered. The most various materials can be used for the latter, including metals (copper, gold, vanadium, etc.), semiconductors (silicon, connections of gallium with arsenide, indium with phosphorus, etc.), and also specific nanomaterials, having both metal and semi-conductor properties (allotropic forms of carbon, nitrogen, phosphorus oxide). Supercompact and superfast elements of computers and communication systems allowing the terahertz frequency range with a significant decrease in total power consumption may arise as a result of the implementation of such materials.
For creation the electronic devices specified above, a comprehensive research of relevant nanomaterials properties and their interaction with the environment within the framework of whole architecture is an uppermost necessity. This problem equally addresses the experimental and theoretical researches. In the latter case the computer modeling, including that one on high-performance computer systems, is playing the growing role. The high computational load is connected with the considerably complicated mathematical models describing processes in the new electronic devices. In this models along with the traditional factors (tridimentionality, non-stationarity, non-linearity, existential instability and an incorrectness of problems) the novel ones, such as the complex real geometry, the whole hierarchy of the existential sizes, non-locality of processes, multi-component media and multi-phase physical values, are considered now. As a result, the modern mathematical models include the whole set of the physical descriptions having the various nature and using various mathematical apparatus (continuum mechanics descriptions, quantum and statistical models, hybrid approaches), as a rule, badly joined together.
Another difficult element of a model is the geometry of computational area and the geometrical computation technique used in calculations [1.1, 1.2]. The problem is that the detailed geometrical configuration of the electronic device should be taken into consideration when the spatial scales of the active elements of the device approach the molecular and nuclear size. For this purpose it is necessary to know the structural parameters of the nanomaterials under consideration, to have the exact notion of the geometrical and spectral characteristics of equilibrium and quasi-equilibrium states of molecules, atoms and their possible combinations. As a rule, in any specific case these data are either inexact, or insufficient to give the satisfactory description of a modeled configuration. As a result, before starting the simulation of the main problem, one should set and solve special geometrical and spectral problems for obtaining the initial data on geometrical structure of material as well as its possible defects and variations. Further in the course solving the main problem it is necessary to recalculate in parallel the dynamical changes of the geometrical data for the control purposes of the basic calculation geometrical and spectral characteristics. The geometrical and spectral information can be a result of the basic calculations also. Thus, the geometrical analysis is an integral and essential part of computing experiment at all its stages.
Another aspect of this problem consists in the fact that the carrying out of geometrical calculations is inefficient and, in some cases, impossible without visualization of the initial, intermediate and final result data. Therefore, the problem of visualization of the geometrical data of various classes is an important part of geometrical modeling [1.3]. Among the primary goals of visualization the special attention is given to the rendering of implicitly defined data, surfaces and bodies in the high dimension spaces and the dynamics of multidimensional objects. An additional problem is the rendering of the multidimensional distributed data obtained as a result of calculations on multiprocessor computing systems with shared memory.
The purpose of the given work is to illustrate the problems arising in geometrical modeling and visualization by an example of some specific problems in modern nanoelectronics and to outline some ways of solving them. And though the problems considered in the work basically concern the one-dimensional and two-dimensional systems, the offered decisions can be generalized to a multidimensional case.
2. Geometrical methods and visualization in nanoelectronics problems
First let's consider the basic classes of nanoelectronics problems which anyhow can be solved by mathematical methods. We will try to arrange them in order of increasing computational complexity.
The first class of problems is connected with computer representation of electronic schemes and systems at macro- and micro- levels. As a rule, these are various drawings and devices schemes describing their design, components and materials, modes of operation, and tables of parameters. The given class of problems is effectively supported by a lot of CAD/CAE systems and databases. Geometrical problems considered here basically concern the graph theory and visualization of geometrical objects.
The second class of problems is connected with calculations of the composition, structure and properties of various substances and materials used in modern electronics. In this class there exist two basic research directions. The first one is the creation and use of comprehensive databases on substance and material properties integrating both experimental and computational data. The geometrical methods used herein, concern basically the digital coding of the geometrical information. The second direction is formed by the quantum and chemical simulators, allowing calculation of the structure and properties of substances and materials. The basis of numerical and geometrical calculations is constituted here by the variation and quantum problems in multi-dimensional spaces. The results of these calculations are also located in databases.
The third class of problems is connected with modeling the operation of specific electronic devices and their parts, as well as the optimization problems of parameters of devices. In our opinion this component is the most numerically consuming one, using practically all methods of computational geometry and visualization.
In the present work we will consider in detail the third class of problems only. Moreover, our consideration will only concern the problems of modeling some physical processes in certain parts of electronic devices. In these narrower frameworks the following traditional geometrical problems are considered.
1. Description of computational area by geometric methods.
2. Construction of computational grids of various type and quality.
3. Approximation of the basic equations on the given grids using the geometrical information.
4. Solving discrete problems arising from grid approximation, using the geometrical information and geometric methods.
5. Geometrical parallelization and solving the problems involved in parallel mode.
6. Use of geometrical models and methods for visualization of initial, intermediate and final of result data.
The number of scientific works in the listed areas cannot be estimated. Therefore we will represent only the results obtained with the active participation of the authors of the article in the last years [2.1-2.13]. In these works practically all aspects of two-dimensional and three-dimensional modeling of continuum mechanics problems on irregular grids have been considered. These works were focused on the implementation of high-performance multiprocessor computing systems. The approaches to the geometrical modeling and visualization developed here are demanded also in nanoelectronics problems. Below we will illustrate the implementation of this and other techniques by concrete examples.
3. Simulation of electron transport in quantum structures
The studying of non-linear electron transport in quantum structures has a lot of applications in modern nanoelectronics. The last period of time is marked by essential progress in study of spontaneous spin polarization effects in non-magnetic quantum size structures. These phenomena induce the important different non-linear effects that are interested for technical applications [3.1-3.6]. The simulation such problems also leans against methods of geometrical modeling.
Here we consider 1D electron transport in a quantum wire built in active region of GaAlAs heterostructure. Our investigations are based on a new mathematical model of non-linear electron transport in the 1D quantum wire with leads. This model was proposed and analysed theoretically in [3.7-3.8]. The model is based on the Hartree-Fock approach and joins both the quantum wire and metallic leads (which are considered as electron reservoirs) to indivisible system. Mathematical description of this model is given by the Schrodinger equations for forward and reverse electron waves for full energy spectrum and Poisson equation for potential of electric field. We introduce also the spontaneous spin polarization of electron waves into the model.
This modified non-local non-linear model was analysed numerically. For this purpose the new numerical method was constructed and tested [3.9-3.10]. The method utilises the finite-difference schemes for discretization of Schrodinger and Poisson equations. The special iterative procedure was developed to overcome the non-linearity of the problem. In order to apply the multiprocessor computer systems for numerical simulations the decomposition on energy co-ordinate was fulfilled. The small parameter prolongation method reduces the computational costs.
Additional problem arising in simulation is the instability of difference solution. It was successfully overcome using the developed method. The electron system can occupy one of several stable or non-stable states. The proposed numerical method gives the opportunity for selection of these states.
The simulation of electron transport in a quantum wire with the help of presented numerical model was directed on the calculations and analysis of equilibrium states of electron system in the wire. In numerical experiments the four types of stationary states of electron system were observed and studied. Other purpose of simulation is the analysis of spontaneous spin polarization processes in non-magnetic nanostructures. Under certain conditions these processes generate the multi-stable response of system to external influence. This phenomenon can be used to construct the super-high frequency circuits for data storage and processing in modern and future computers.
The propagation and
interaction of electron waves in a quantum wire in the stationary case can be
described with the help of Hartree-Fock approach. In this case of axial
symmetric wire with finite length L and radius a and one-zone
approximation of electron interaction we can use the quasi-1D model [3.7]. This
model consists of the stationary Schrodinger equations for wave functions 
of electrons. These
wave functions describe forward ( 
) and reverse ( 
) electron flows in wire and its are characterized by two
spin values 
. The wave index k runs the range 
(here kF is the Fermi limit). As a
result the problem has two coordinates: spatial (x) and wave (k).
The Schrodinger equation system is closed by the Poisson equation for the self-consistent electric field. In the case of axial symmetry of wire with the help of Green function method we can reduce this Poisson equation to the integral formula for the potential. This formula is included in the wave equations directly. As a result we have the following dimensionless equations
(3.1)
Here spatial co-ordinate x and inverse value
of wave index k are normalized on the
wire length L, 
and 
are the electron wave
energies, the parameter 
is the average value
of potential energy of electron system as a whole.
In the system (3.1) term
(3.2)
describes the potential
energy. This energy has the following components. The first term 
is an effective value
of potential barrier built in active region of heterostructure. The second term
describes the spin correction to potential energy for different electron waves.
The coefficient b is the factor of spontaneous
spin polarization. The term 
is an applied voltage
function having the maximum value V.
The term 
is the Hartree energy,
,
(3.3)
where 
is the positively charged
background density, 
is the electron density,
(3.4)
In (3.3) we assume for simplicity that the radial component of the background charge density is the same as the electron density. And in (3.4) we use the step electron distribution function (independent of temperature).
The electron-electron
interaction potential 
that appears in (3.3)
in our case has the simple form
(3.5)
The last term in (3.2) 
is the exchange energy
operator,
(3.6)
where
(3.7)
In a finite wire the boundary conditions for the wave functions can be reduce to the following form
(3.8)
where 
, 
.
For numerical simulation of
the problem (3.1)-(3.9) we used the energetic co-ordinate 
instead of wave co-ordinate 
. As the result we have the following formula
(3.9)
where 
is any function from (or ).
The numerical method elaborated for the problem (3.1)-(3.9) is based on the approaches that are described in [3.9-3.10]. Let us describe the main steps of the method.
For discretization of
computational domain 
we used grid 
with components 
, 
. The difference analogies 
of wave functions were
defined on the grid 
. For the discretization of equations (3.1) with boundary
conditions (3.9) the conservative monotonic finite difference schemes were
constructed with the help of finite volume method. As a result the following
finite difference equations were derived for the calculation of electron wave
functions:
(3.10)
where
(3.11)
The term 
has the following
structure
. (3.12)
Note that the equation system
(3.10)-(3.12) is strongly non-linear. The non-linearity is induced by the
Hartree energy operator 
and operators 
of electron exchange
interaction. This non-linearity has a non-local character since the terms
depend on all wave functions. For overcoming this difficulty the following
iterative procedure is proposed.
Let us consider the
combinations 
as a joint non-linear
difference operators 
which acts on each
forward wave function 
. Analogously, the operators 
acts on each reverse
wave function 
. In the both cases the operators 
depend on a full set
of wave functions 
. The set 
depends on the
operators implicitly. These
facts can be expressed by the operator equation system
(3.13)
We can use non-linear system
(3.13) to solve the difference problem for each fixed value 
of applied voltage.
For this we organise the following iterative process
(3.14)
Here the initial operators 
are equal to
null-operator and the iterative parameters 
depend on the spectrum
of operator functions 
. The details of this numerical algorithm and its parallel realization are
resulted in the works [3.11, 3.12].
The numerical algorithm presented above is based on the net approach and uses geometrical calculations. However application of geometrical methods in the following situation is more essential. The matter is that the put differential problem and its discrete analogue under some conditions are not correct. In particular, if on external electrodes of a quantum wire to submit pressure electronic waves can pass through it in several ways. That is stationary electron transport through a quantum wire will be unstable, namely, bi - or multistable. It means that the put differential problem (together with разностная a problem) has some decisions. One of possible ways of the permission of the given difficulty is application of a geometrical method. Let’s consider the details of this method.
For one-dimensional systems in which the electric current proceeds, it is known, that their current-voltage characteristic (CVC) is a continuous curve. Therefore, even in bi-stable or multi-stable systems it does not lose this property and only contains sites S or N type (see Figure 3.1a). Each point on the CVC corresponds to a steady or unstable branch of the decision. And to find all decisions for a field or current preset value, it is enough to spend loading lines through all branches of the CVC and to localize crossing points. For a known CVC it becomes evidently.
If the CVC is unknown (it is defined together with the problem decision), then other approach is applied. In the field of instability the additional equation describing a curve of loading and allocating the unique decision is added. For this purpose only it is necessary, that the loading curve was crossed with CVC in one point. To provide this condition on full plane it is practically impossible. However, if to allocate a small vicinity near CVC, the giving of curve loading will be possible.
In a method offered in works [3.13, 3.14] the following procedure was used. The CVC is described by the formal equation
.
(3.15)
It is computed from a balance point (j0=0, E0=0). Usually this condition of system is unique. Its calculation does not represent the big difficulty. The initial site of CVC in a vicinity of a point of balance is close to linear function. Therefore at a small step on both co-ordinates (j, E) it is easy to find the following some points (jk, Ek). Further in last calculated point (jk, Ek) it is placed local system of polar co-ordinates (ρ, φ). For corner readout φ we choose a tangent to the CVC in a point (jk, Ek). For calculation of a new point of the CVC (jk+1, Ek+1) we will write down in a point (jk, Ek) the equation of curve loading. One of the convenient equations of curve loading is the equation of the Witch of Agnesi:
,
(3.16)
where 
and 
are some parameters which sense is clear from Figure 3.1b. This curve is close to a circle, but
unlike last, it crosses the CVC only in one point if the radius is small enough. The joint decision of the equations (3.15) and (3.16) gives a new point of
a curve CVC (jk+1,Ek+1).
(a) |
(b) |
Figure 3.1. Example of instable non-linear current-voltage curve (black color)
and loading lines (blue and red colors) (a) and geometrical method illustration (b).
The green points mark the different decision branches.
Proposed geometrical method was applied to problem (3.1)-(3.9). The results of numerical simulation are observed in [3.11, 3.12]. Here we will discuss the numerical experiments connected with computation of current-voltage characteristics.
The Figure 3.2 presents the
current-voltage characteristics of wire obtained for different values of spin
polarization factor 
. The analysis of presented data allows formulate the
following statement. The spontaneous spin polarization leads to instability of
electron system in the wire and resulting current-voltage characteristics can
be strongly non-linear with the multivalence parts. This fact confirms that the
electron system demonstrates the amplification or multistability properties
under developed spontaneous spin polarization.
Figure 3.2. Current-voltage characteristics 
for different values
of spin polarization factor 
(curves 1-5).
We observed several aspects of the steady-state electron transport in a quantum wire. The main direction of this analysis was the influence of spontaneous spin polarization on the parameters of electron system in both equilibrium and non-equilibrium states. The numerical results allow us to formulate the following hypothesis. Some special control for the spontaneous spin polarization can be used for realization of super-high frequency quantum transistors and multi-logic elements. For example, this control can be realized by small correction of the magnetic field near a quantum wire. These results have been received thanks to sharing of methods of numerical and geometrical modeling.
4. Simulation of electron transport in vacuum micro- and nanostructures
Semiconductor submicron structures and nanostructures are now the basis of modern solid state electronics. Numerical simulation of their static and dynamic characteristic is an actual practical problem and adequate mathematical modeling of electron transport in these structures based on clear understanding of its fundamental physical peculiarities is vital for successful development of submicron devices.
The size of an active space in such micro-devices is comparable to a charge carrier characteristic free path and the electric field in the space due to its small size is very high for actual bias. For that reason electron transport in active domain is usually quite non-equilibrium one and widely used drift-diffusion model, DDM [4.1], fails to be adequate. The quasi-hydrodynamic approach to strongly non-equilibrium electron transport in semiconductor is more correct.
Quasi-hydrodynamic model is physically based on inequalities [4.2]
,
(4.1)
between the characteristic
times of free charge carrier system in the active space of semiconductor
structure. Here 
, 
are momentum and
energy relaxation times respectively, and 
is electron-electron
scattering rate. The inequalities (4.1) results in almost maxwellian form (or
Fermi-type) of the symmetrical part of charge carriers distribution function
with non-equilibrium temperature 
. That enables in case of an elastic momentum scattering to
describe the anti-symmetrical part of the distribution function in terms of two
variables: local charge carrier density and their local temperature - e.g.
electron concentration 
and electron
temperature 
for unipolar electron
semiconductor. The latter obey the
continuity equation and the energy balance equation closed with the material
equations for the current and energy flux with concentration and temperature
included into the corresponding kinetic coefficients.
In the conditions of strong
electron heating the process of impact ionization becomes very important. In
this case hole conductivity is strongly different from electron conductivity,
and the semiconductor exhibits bipolar properties. In this situation the
continuity equation for hole concentration 
is added to the model.
The Poisson equation for self-consistent electric field 
and corresponding
boundary and initial conditions close this model.
The resulting 3D evolutionary differential problem is strongly nonlinear one. The non-linearity is taken place both in equations and boundary conditions. The numerical analysis of the problem in quasi-classical approximation and possible applications to silicon field emitter simulation are considered in the work.
In general case the discussed differential problem can be written by following dimensionless quasi-classical deterministic equations:
(4.2)
(4.3)
(4.4)
(4.5)
where
Here 
is the local
electron energy density, 
, 
are electron and hole current densities, 
and 
are generation and
recombination terms, 
is the electron energy
flux, 
and 
are generation and
relaxation energy rates, 
is the electric field
potential, ND is the
effective donor concentration. The 
, 
, and 
, 
are the kinetic
coefficients for the corresponding transport equations: 
, 
are respectively the
electron and hole mobility, 
, 
are the diffusion
coefficients, c is the Peltier coefficient.
The constant 
is static dielectric
permittivity of the material. The 
, 
are standard
differential operators determined in the domain , time 
.
The boundary and initial conditions can be written in form
(4.6)
(4.7)
(4.8)
where 
is the domain bound, 
is the normal to , 
and 
are equilibrium distribution of electron and hole
concentrations, 
is the environment
temperature.
In works [4.3-4.8] the quasi-hydrodynamic approach was applied to modeling of field emission from semiconductor micro-cathode. This problem is connected with following.
In recent years vacuum microelectronics (VM) based on field emitter array [4.9] (FEA) concept has experienced tremendous growth (see Figure 4.1). VM devices benefit consists in faster modulation and higher electron energies. These properties are achieved by using solid-state structures. In addition VM devices can operate in a wide temperature range, 4K<T<1000K, and in high-radiation environment. Applications include FEA flat-panel displays, microwave amplifiers, digital IC, electron and ion guns, sensors, high energy accelerators, electron beam lithography, free electron lasers, and electron microscopes and microprobes. Innovative approaches to FEA’s were made in latest years with respect to FEA materials, structure, and applications. A number of VM devices have by now moved beyond the research laboratories to actual prototypes and commercial products.
|
|
Figure 4.1. Example of silicon FEA (left) and one silicon edge cathode (right).
Now silicon is considered as one of the most suitable material for FEA fabricating in batch technology (see, e.g. [4.10]). Silicon, though has orders of magnitude fewer conduction band electrons than metals, has emitting characteristics comparable to metals, and highly developed silicon batch technology can be applied to make various VM devices, including transistor-like structures.
There are many specific aspects and special requirements for VM, and comprehension of physics of VM devices functioning play a key role in its successful development. One of the most important problem is the creation of FEA with sufficiently high, controlled, and stable emission ability. Field emission from semiconductors has some peculiar features vital for successful development of VM devices.
High electric field penetrates deep enough into the semiconductor and results in intense electron heating near the emitting surface. Since the tunneling coefficient exponentially depends on energy, this drastically affects the emission characteristics and heat dissipation. Thus the electron transport in semiconductor field emitter is in fact highly non-equilibrium hot electron process. It was well demonstrated in [4.3, 4.4] for one-dimensional model. Electrons with the energy higher than the semiconductor band gap contribute essentially to the emission current. So impact ionization can play an important role and should be taken into account. Besides, the quasi-electrostatic problem in a microcell space outside the cathode must be solved self-consistently to obtain current-voltage characteristics.
For the above reasons multidimensional (2D/3D) approach is principal for real cathodes microcell modeling. It is an actual physical and practical problem that is rather complicated and can be solved accurately enough by means of numerical methods only.
The numerical realization of
proposed difference problem (4.2)-(4.8) is very complex computational problem. In
works [4.3-4.8] the simple geometry case when is 2D rectangular
domain was analyzed. In this case to solve the problem (4.2)-(4.8) the original
additive difference method proposed in [4.11] was applied. The results of
simulations in this case were proposed in [4.8].
The offered approach has received the further development in works [4.12-4.15]. In these works the basic attention has been given real two-dimensional geometry of a problem and parallelism. Now the geometrical modeling became the central part of calculations. For decision of the problem the following steps are needed:
1. Construction of geometry editor for setting of real structures.
2. Developing of triangular grids generator.
3. Approximation of basic equations on irregular triangular grids.
4. Construction of program for rational domain partitioning.
5. Developing of parallel algorithms and programs for decision of basic equations with the help of multiprocessor computer systems with distributed architecture.
The main part of these tasks is based on geometrical computations.
For decision of first tasks the program tool TrgView was constructed. This tool allows both to build and to visualize the different 2D structures including of different materials (see Figure 4.2). This tool allows see also a triangular grid ( Figure 4.3) and its splitting into domains ( Figure 4.4).
For achievement of the second and fourth purposes the program complex MeshGen of generation of various triangular grids has been created. The results of its work can be see on Figure 4.3, 4.4. The algorithms of mesh generation were published in works [4.16-4.18]. The domain decomposition methods were developed in [4.19, 4.20].
|
|
Figure 4.2. Diode (left) and triode (right) structures of silicon microcell. Domain Ω1 is silicon, Ω2 is vacuum, Ω3 is insulator SiO2.
|
|
Figure 4.3. Visualization of structure (left) and grid (right) in program tool TrgView.
Figure 4.4. Visualization of domain partitioning.
The task of approximation of basic equations on irregular triangular grids was considered in works [4.21-4.25]. With the help of finite volume approach the new numerical methods were constructed and analyzed. The median control volumes were used for approximation on triangular grids. On Figure 4.5 the examples of control volumes are shown.
|
|
Figure 4.5. Examples of control volumes.
To solve transport equations
(4.2)-(4.4) the developed finite volume schemes were used. These schemes guarantee
the conservativity and weak monotonisity of the numerical solution. For grid
analogue of the Poisson equation with discontinuous dielectric permittivity
coefficient the same approximation
was used. For the decision of non-linear finite difference schemes the uniform
external iterative process were elaborated. The finite volume flow equations
and Poisson equations are solved with the help of independent internal
iterative procedures based on Cholessky factorizations and conjugate gradient
method. The parallel variants of these methods were developed in works [4.26,
4.27].
The visualization of large distributed grid data has been executed by means of program RemoteView that was developed in [4.28-4.30]. The visualization of small grid data has been executed by means of known program TecPlot.
The some results of numerical simulation are given on Figure 4.6-4.8. Its show that electric field deeply gets deep into the semiconductor. This fact is the main reason of degradation of the device. In our calculations the critical conditions of electron emission and current-voltage characteristics of devices were obtained. For more details see [4.15].
Figure 4.6. Distribution of electric field modulus in silicon edge micro-cathode.
Figure 4.7. Distribution of electron temperature in silicon edge micro-cathode.
Figure 4.8. Distribution of electron energy in silicon edge micro-cathode.
In final of this section can be carried out that geometrical modeling help to solve large set of complex problems in this field. Future of this work will be directed on full 3D parallel simulations. The problem of dynamic tunneling of electrons through a potential barrier will dare thus. The first step of this work was executed in [4.31].
5. Simulation of nucleation and migration of voids in interconnects of electrical circuits
Problems of void nucleation in metals at various physical influences are known for a long time. With occurrence and development of electronic devices processes of void nucleation in metal connections of electric schemes of a steel by one of the basic obstacles in a way of working out of failure-safe systems [5.1]. In modern electronics of a problem of void nucleation remains actual in connection with transition to manufacture of devices in nanometer range. At such sizes even insignificant defects of materials very quickly lead to destruction, both separate schemes, and the whole chips. Therefore manufacturers of microcircuits are compelled to develop new approaches to the decision of the given problem. The last is impossible without carrying out of detailed researches of processes of formation of a time and their migration on spending lines of microcircuits [5.1-5.8].
In the work [5.8] one of possible approaches to simulation of processes of nucleation and migration of voids in interconnections of electric schemes is offered. For this purpose the nonlinear mathematical model of void nucleation processes is developed. This model includes the stationary equations for electric field, carrying over of heat and elastic pressure in the layered environment containing an interconnection. Also it includes the non-stationary equation of diffusion for concentration of atoms of metal in a line and an interconnection.
For an example the problem is considered in flat geometry. As a line and interconnection material copper with a tantalic lining on an external contour acts (see Figure 5.1). For the chosen statement of a problem schemes are constructed monotonous conservative finite difference schemes. The equations for potential of electric field, temperature and a component of a vector of displacement are elliptic and are realized by means of iterations. The non-stationary equation of diffusion dares under the obvious scheme. Calculations are spent by means of a complex of parallel programs.
The given problem has a close connection with methods of geometrical modeling. The matter is that void nucleation process represents phase transition: substance - vacuum. The mathematical description of this phenomenon is carried out by a geometrical method, namely, introduction in model of order parameter in which quality the copper mass fraction acts. The given technique is in details developed in book [5.7]. This technique consists that the system of the differential or integrated equations describing process includes linear or nonlinear dependences on order parameter as factors. These dependences lead to spasmodic change to behavior of the required decision at change of a geometrical configuration of model. For more details see [5.7].
Let's consider now is short mathematical statement of a problem and some results of modeling.
Figure 5.1. Problem geometry. Both digits (1, 2, 3, 4) and colors correspond to structure materials: copper (1), tantalum (2), silicon carbide (3) and insulator (4). The rights show the electric current direction.
The processes of nucleation and migration of voids can be described by follow basic un-dimensional equations
(5.1)
(5.2)
(5.3)
(5.4)
Here 
, 
, 
and are current density, non-linear
conductance, electric field strength and potential; 
and are divergence and gradients operators
in Cartesian co-ordinates 
; 
is calculation domain with linear
sizes 
, 
; 
, 
, 
are thermal flow, thermal conductivity
coefficient, temperature of area, 
is un-dimensional parameter; 
is scalar product in space of ; 
are vectors that consist of thermal
stress tensor 
, 
is displacement vector; 
, 
and 
are Lame constants and compression
modulus, 
is stress function that describes
thermal expansion and mass changes; 
is mass fraction of copper, 
is time, 
is total diffusion flow, 
and 
are non-linear diffusion
coefficients, 
is generalized thermodynamic
potential ( 
is trace of tensor ), 
is sub-domain of occupied of copper and voids.
The boundary and initial conditions are following
(5.5)
(5.6)
(5.7)
(5.8)
Here 
are current densities on
metallic lateral faces, 
is total current, 
is chip temperature, 
is equilibrium mass
fraction of copper.
The coefficients
of equations (5.1) – (5.4) are discontinuous. They depend on temperature and mass
fraction of copper non-linearly. The concrete form of these coefficients is
determinate of electric circuit parameters. The thermodynamic potential 
has the
following form
,
(5.9)
where 
are some numerical constants, 
is
For an illustration of efficiency of the used
geometrical approach we will consider some results of simulation. Let's
consider at first electro-migration process in geometrically symmetric element
of a spending line (see Fig 5.2). The chosen fragment of a line has length 8
microns, a thickness 0.6 microns, at line edges there is a layer of tantalum in
the thickness 0.1 microns from each party. In the presented configuration at
the left the copper layer is bared, on the right it is closed by tantalum. It
is supposed, that during the initial moment the fragment of a line has no
defects, that is the quantity of vacancies and atoms of copper are equal to the
equilibrium values in all volume of a line. For an example the equilibrium mass
fraction of copper is equal in the test 0.98. We assume, that the electric
current joins instantly. The current size is equal 
pА, that corresponds to conditions
of testing of chips. The line temperature at testing makes 628 K, the
temperature of fusion of copper lays close 673 K. The length of diffusion
numerical experiment relied equal 10 nm, that corresponds to experimental data.
Accordingly, it is width of an interface between tantalum and copper where a
time arises.
Figure 5.2. The symmetric fragment of electric line. Blue and red colors correspond to copper and tantalum fractions. Hereinafter the sizes are specified in microns.
Calculation results of evolution of a mass fraction of copper near to legal edge of a line are presented on Figure 5.3. These results show that under the influence of an electric current non-equilibrium vacancies of copper atoms of are accumulated in corners of the right part of a line and two voids are formed ( Figure 5.3b). Further this voids start to grow towards each other along a tantalum layer ( Figure 5.3c-5.3e). As a result they unite during one big void (рис. 5.3f). This void blocks a way to an electric current in copper.
The distributions of current density modulus at start of void nucleation and after its formation are shown on Figure 5.4. At begin of void nucleation the current near to a line right edge flows as on tantalum (a small part), and on copper layer (the basic part). After void nucleation the way to an electric current on copper is gradually blocked. As a result the current near to a line right edge flows only on tantalum ( Figure 5.4b). The resistance of tantalum on one and a half order above, than at copper. Therefore the integrated resistance of a line is increased essentially. This effect leads to negative consequences (misoperation of the chip or its full exit out of operation).
Let's consider further results of calculations for more complex geometrical configuration ( Figure 5.1). In this scheme resulted in drawing most a weak spot an interconnection of two lines (on Figure 5.1 the interconnection is placed in center of scheme). As the interconnection has a small thickness on it the maximum current flows. On Figure 5.5, 5.6 the distributions of copper mass fraction and current density modulus are shown for both cases: before void nucleation and after of it evolution. As well as in the first test, it is void blocks the basic channel of passage of a current that it is seen on Figure 5.6b. At the left and to the right of an interconnection in tantalum two cords of a current which can potentially destroy an interconnection were formed.
In conclusion of the given section we will underline, that the geometrical approach to modelling of phase transitions is rather effective. With its help it is possible to study both destruction of electronic systems, and formation of new connections and materials for nanoelectronics.
(a) |
(b) |
(c) |
(d) |
(e) |
(f) |
Figure 5.3. Distributions of copper mass fraction for 
seconds.
(а) |
(b) |
Figure 5.4. Distribution of current modulus before void nucleation (а) and after void evolution (b).
(а) |
(b) |
Figure 5.5. Distribution of copper mass fraction in interconnection before void nucleation (а) and after void evolution (b).
(a) |
(b) |
Figure 5.6. Distribution of current density modulus in interconnection before void nucleation (а) and after void evolution (b).
We have considered the problem on a role of geometrical methods and visualization at modeling of nanoelectronics problems. We had been formulated the basic directions of application of geometrical methods in problems of the given class. On concrete examples efficiency of using of geometrical methods has been shown. The structural modeling not considered by us uses the technique of geometrical methods in even large volume. Orientation to supercomputer geometrical calculations and distributed visualization of large volume data should give in the future still a greater prize in efficiency of the decision of nanoelectronics problems.
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