the design of
carrier domain devices
for non-linear signal
processing
canu
K,
toe aesiign
)Ï domain devices
■ non-lïnear signal
cessinq
Proefschriftter verkrijging v3n de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Delft, op gezag van de Rector Magnificus prof.dr. J . H . Dirken voor een commissie aangewezen door het College van Dekanen te verdedigen op dinsdag 18 juni 1985 te 14.00 uur
door
A.C. van der Woerd
elektrotechnisch ingenieur geboren te Leiden
rov. ' Din1 ï
TR diss
1444
Dit proefschrift is goedgekeurd door de promotor prof.dr.ir. J . Davidse
CHAPTER I GENERAL 1 1-1 General introduction I 1-2 Various considerations concerning the design of carrier
domain devices 6 1-3 Some basic CDD-configurations for non-1inear signal
transfers 9 1-3-1 Introduction 9 1-3-2 CDD's with curved base/emitter geometry, linear
col lector contacts and symmetrically" positioned bias
inj ection contacts 10 1-3-3 CDD's with curved base/emitter geometry, symmetrically
posit ioned bias inj ection contacts and annular buried layer 1 ! 1-3-4 Basic CDD-configurations with asymmetrically
positioned bias injection contacts 12
CHAPTER II THE MODELING OF VARIOUS BASIC CDD-CONFIGURATIONS 15
II-1 General 15 I1-2 The modeling of some basic configurations selected earlier ]5
11-3 Straightforward analysis of the simplified models 17
II-3-1 General 17 11-3-2 Straightforward analysis of the collector area or
buried layers including their contacts 17 II-3-3 Mathematical considerations of the base-emitter
geometries 20 II-3-4 The model of Figure 2-2b 20
I1-3-5 The model of Figure 2-2d 21
II-3-6 The models of Figures 2-2a and 2-2c 22
II-4 Optimal shape of the bias injection contacts 27
CHAPTER III ITERATIVE ANALYTIC SYSTEMS TOR DESIGNING CDD'S WITH
ACCURATE NON-LINEAR TRANSFER FUNCTIONS 31
III-1 Introduction 31 III-2 The design of CDD's with v.ariable emitter locus and
I I
III—3 Restrictions which have to be imposed on the functions to be realized
111-4 The design of CDD's with variable emitter locus and asymmetricallv positioned central inj ection contact III—5 The application of the extended iterative system with
two different collector geometries
111 - b 1' h e analysis concerning the mir to ring models
III—6-1 Introduction
111-6-2 Model ing with conductive paper 111-6-3 Conformal mapping
III-b-4 Finite-difference and finite-elements methods III-6-5 Calculating the potentials along the boundary of" a CDD base-disk with the finite difference me thod
III—6—6 Green's boundary formula
III-7 Calculating irregularly shaped planar resistors by Green ' s boundary formula
III—& Comparison of the described methods and conclusions
CHAPTER IV CARRIER-DOMAIN DEVICES IN PRACTICE IV-1 Introduction
IV-2 The effect of a few important i.e. process parameters o the quality of CDD's
IV-2-] Introduction IV-2-2 Low-frequency behavior IV-2-3 high-frequency behavior
IV-2-A Some remarks on the process choice
IV-3 The design of an accurate wide-band analog multiplier IV-3-1 Two early design examples
1V-3-2 An optimally designed CDD-analog multiplier IV-4 An accurate wide-band triangle-sine converter IV-5 Justification of the choice of the base/collector
structure
IV-6 Design examples of CDD's with relatively large values of the first derivative of the function
IV-7 Biasing conditions for carrier-domain devices IV-7-1 Introduction
IV-7-2 The minimum supply voltage IV-7-3 Emitter current
IV-7-4 Bias driving currents 77 IV-7-5 Drive and load circuit 78 IV-8 The effect of mask alignment errors and inhomogeneities
of the resistive layers 79 IV-Ö-] Introduction 79 IV-8-2 horizontal translation of the base/emitter geometry
with respect to the collector contact diffusion 81 IV-8-J Vertical translation of the base/emitter geometry
with respect to the collector contact diffusion 82 IV-8-4 Rotation of the collector contacts with respect
to the base/emitter geometry including the injection
contacts 82 IV-8-5 Horizontal translation of the bias injection
contacts with respect to the base/emitter location 83 IV-8-6 Vertical translation of the bias injection
contacts with respect to the base/emitter location 84
IV-8-7 Inhomogeneities 34 IV-8-8 Spread measurements 86
IV-9 Noise in CDD's 89 IV-IÜ DÏscussion 90
CHAPTER V CDD'S AS MULTIPLE COMPARATORS AND ANALOG SWITCHES 92
V-] Introduction 92 V-2 Multiple comparators for A/D-conversion 93
V-2-1 Introduction 93 V-2-2 Choice of the basic configuration 93
V-2-3 Some notational conventions 95 V-2-4 The angular positions of the collector contacts 96
V-2-5 Detection principle 96 V-2-6 Detection criterion 97 V-2-7 Two early devices 98 V-2-8 Improving thedetectibility 100
V-2-9 Improving the collector discrimination 102
V-2-I 0 Dummy collectors 105 V-2-1] Calculation of the detectibility 106
V-2-1 2 Conclusions n o V-3 Analog switches with CDD's n O
V-3-! Introduction ]i0 V-3-2 Cross-talk 1 ]Q
IV Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F References Summary Samenvatting A-1 B-l c-i D-1 E-l F-l G-l H-l 1-1 CHAPTER 1 GENERAL 1-1 General introduction
With Barry Gilbert's invention of "resistive conversion devices" [1-1] a new branch was added to the already richly variegated family tree of semiconductor components. Over time, these resistive conversion devices have got ten the name "carriet domain devices" (CDD's). As the name indicates, the common property of these devices is the presence of a group of carriers, if only this property were considered, however, any semiconductor device would be a CDU. The most salient feature of a CDD is that the group of carriers (the domain) can be laterally displaced over some bounded semiconductor area by means of independent external parameters. Almost without exception, the contraction of the carriers, in other words the creation of the domain, is accomplished by a field being deliberately applied to the base area of a bipolar transistor. In common bipolar transistors a similar phenomenon, known as emitter crowding, occurs, and generally has a negative effect on the device properties [1-2]. In a CDD, however, this effect is intentionally brought about by inserting external currents somewhere in the base area.
The main feature of CDD's is that the position of the domain can be
controlled by external currents. This can best be illustrated by a simple
example. Figure 1-1 depicts a simple ftPN transistor which can be construct ed in a bipolar standard process. The base area contains three contacts into which the currents -(] + X)I , 21 and -{I - X)I which can easily be obtained from a balanced transconductance amplifier, are inj ected. The parameter X is controlled by the input voltage V. and equals ± I if
I 'R . vo kT V = + _£~—E (assuming R_ » —-I y where V is the thermal voltage — ) .
If X = +1, the base-emitter voltage on the left-hand side of the junction will be negative with respect to that on the right-hand side. If this voltage difference exceeds a certain value, conduction is only possible on the right-hand side of the emitter. If X = - 1 , the polarity changes and only the left-hand side of the junction can conduct. If, finally, X = 0 the conduction domain will lie exactly in the center of the emitter. The result is a kind of continuously "walking" transistor: it can be regarded as a voltage-to-position converter.
Isolation diffusion Emitter
Base-collector layer
Figure 1-1. NPN-transistor structure with three base contacts and. driving cireuitry.
The. device of Figure 1-1 is of no practical value unless we are able to
detect the position of the conducting domain. As the emitter current flows
towards the collector it is possible to detect where the conduction is taking place by providing the transistor device with more than one collector contact. The detect ion can be carried out either discretely or
continuously.
Figure 1-2a and b depict extensions to Figure 1 -1. In Figure 1-2a there are two collector contacts. The inj ected current will be divided up between these two contacts accurding tu a resistive law. Thus the difference between these two partial collector currents is a continuous measure of the position of the current domain. Hence, this current difference will have a (non)linear relation to the input variable X.
collector contacts Figure l-2a collector contacts emitter base-collector layer Injection contacts Figure l-2b
Figure 1-2. Two versions of the device shown in Figure 1-1, extended with collector contacts.
In order to normalize the resulting (non)linear function we introduce
the common-base current-amplification factor, Y has the same interval as X, i.e. the interval [-1, +1], Hence, the normalized (non)linear transfer function is now Y = f (X) . This form of notation and normalization will be maintained throughout the remainder of this work.
Closer inspection of the principle shown in Figure 1-2a reveals that this principle offers the possibility to design CDD's with certain pre~ aetermined non-linear transfer functions. What strikes the eyes is a
particular feature, namely that the accuracy of the transfer function will be determined by geometrical aspects rather than by electrical
aspects. Hence the accuracy of a CDD will always depend on the accuracy of the masking and edging methods of the i.e. process employed. Further, because the position of the domain and its detection process depend on local sheet resistivities, the homogeneity of the different layers will play an important role.
Configurations based on the principle shown in Figure 1-2b are suitable for various other interesting purposes. For different discrete values of X the emitter current will flow to one of the collector contacts. The result is an electronic analogon of a rotary switch controlled by the input voltage. If the emitter current is kept constant and V is chosen to be a continuous variable, another application results, namely a plural comparator that can be used for parallel A-D-converters.
Some definitions
A few terms which will crop up again and again in this work are defined below:
i) Domain width
Figure ]-3 depicts a possible potential course along the base-emitter junction of a geometry like that of Figure 1-1. It has been drawn for X = 0 and X - ± 1 .
level of emitter voltage
Figure 1-3. Possible potential course along the base-emitter path,
The local emitter current density can be calculated at any place along the junction. If we assume that the potential along the junction is described by the function V_ = V(x), then the local emitter density is
J(x) = J exp V(x) (1-1)
where V is the thermal voltage kT/q and J a certain normalizing constant. In order to talk sensibly about the domain operation of a CDD, an expression which gives the width of the conduction domain is indispensible. An arbitrary, though convenient measure is found if we define the domain width as the distance between the two points where the emitter current density has decreased to 60% of its maximal value. The potentials in those
1 kT
points are then -^ — lower than the potential maximum. If the base-emitter
shape constitutes a half-circular disk such as in Figure 1-4 (this geometry occurs a few times in subsequent chapters), the exact value of the domain width according to the given definition is [1-3]
23. 14
(1-2)
where p is the sheet resistance of the base layer and I is the average bias current. In other cases the domain width has to be numerically calculated.
ii) Domain centroid versus potential maximum.
If the product Pp^p were infinite, conduction of the base-emitter junction
« F ®
emitter
SEIMI-cincular base
f a
-could only occur in the potential maximum. In practical cases this assumpt ion is not allowed. In such cases a relatively large part of the j unction contributes to the device operation. The contributions of all points in the junction can be replaced by one point, called the domain centroid, which acts as a kind of "center of gravitation", If its posit ion is properly chosen, the ultimate operation of the device remains unaltered. Its position has to be calculated in every individual case by means of numerical methods.
To conclude this section we present a brief outline of the contents of this work.
In Chapter I, some general design considerations and a classification of basic CDD-configurations are presented. A design method is given which yields a simplified model directly suitable for (inaccurate) designs.
This model is compared with a more accurate one that can be anaLyzed by
computer. A suitable interaction between design and analyses leads to accurately operating devices.
Chapter II contains a simplified and more accurate modeling of the basic configurations, together with some analyses concerning the simplified models.
In chapter III some computer-aided design methods based on the models and analyses of Chapter II are given. Next, some practical devices are dealt with in Chapter IV.
Chapter V is somewhat distinct, as it treats the analyses and development of some multiple comparators which operate according to the carrier-domain principle. Some practical examples and their applications are given as well.
1-2 Various considerations concerning the design of carrier-domain devices.
In designing a carrier-domain device, the designer may make a choise out of some basically different approaches. Which strategy is chosen depends on the designer's knowledge, the desired action of the device, Integrated-circuit processes and software which are available and, last but not least, the designer's personal preferences and inventiveness.
-7-A design strategy which is frequently observed is the following one. The designer finds out more or less by chance that some geometry produces a certain usable transfer function. Then, in order to gain more insight into the primairy and secondary properties of the device, he or she tries to simulate the device by means of a mode], which is accessible and subject to manipulation by mathematical methods. As soon as it appears that the model is inadequate for the actual device (which can be at tributed to second-order effects which cannot be included in the mathematical expressions belonging to the model) lie or she attempts to adapt the actual device so as
to improve its correspondence to the chosen mode 1.
Such an approach was used by Smith and Mainley for their analog CDD-multipliers [1-4] , [1-5]. It should be emphasized that this strategy does
-not imply a real design method^ for one starts with a certain geometry and
not with the final objective. Hence, a free choice of the ultimate nature
of the operation of the device is out of the question.
An early design method, where the transfer function of the device could be chosen freely, has earlier been introduced by the author of this thesis [ 1-6]. Since it is essential to the later discussion of more advanced methods,
it will be briefly discussed below (More details and a few early experiments will be discussed in Section
II—3)-First, the required transfer function is defined and then a basic CDD configuration is chosen. All except one of the geometrical aspects of this configuration are nonvariable. Further, the basic configuration is modeled in such a way that the geometry of the variable part can be calculated. Finally, the device is constructed according to the basic configuration, which is now complete. Unfortunately, it is only possible to calculate the shape of the variable part if the models are greatly simplified. Consequent ly, devices complying with these models fall short of the desired specific ations .
Though accurately calculating the geometry of a device with a predetermined transfer function is generally impossible, the opposite operation (i.e.
the calculation of the transfer function produced by a device with a certain geometry) is basically always possible, whatever the complexity
of the model is. This complexity can only be restricted by the acceptable level of complexity of the required software or by the necessary amount of computing time. Unfortunately, the latter operation is not directly suitable for design purposes. Only after numerous calculations with mutual ly slightly different geometries would it be possible to discover the desired connections between certain wished-for transfers and the proper geometries.
However, the method is very suitable for accurate analyzing. Combining this
method with the previously discussed (inaccurate) design method makes an accurate design method possible. Indeed, the development of such methods
is a paramount objective of this work. The basic principle will briefly be explained with the aid of Figure 1-5. A detailed description will be given in Chapter III.
MIRRORING MODEL error compensation SIMPLIFIED MODEL (^-design
Fig. 1- The introduction of a mirroring model.
After the required transfer function is defined, a suitable basic configuration is chosen. All except one or two of the geometrical properties of this configuration are defined. Two models of the same basic configuration are used. First, we construct a model that is sim plified to such an extent that direct design of (inaccurately operating) devices is possible. As the other model corresponds as closely as possible to the basic configuration, it is called the "mirroring" model. It is introduced for accurately analyzing the action of the device. Both models take part in an iterative process. The variable part(s) of both models
-9-are equal at every stage of the process. During the process any errors which are caused by the inadequacy of the simplified model are analyzed by calculation with the mirroring model. These errors are compensated by adapting the variable part(s) of both models. The process is stopped as soon as the accuracy exceeds a certain predetermined value. Not only can a large number of accurate transfer functions be realized, but other propert ies such as efficiency and power consumption can be optimized as well. Some attendant practical advantages of this CAD-method are formulated below.
a) Tlie requirements on the driving circuitry imposed by the devices can be lowered considerably.
b) As the total required power consumption of the CDD is minimized, more accompanying circuitry can be placed along with the CDD on one chip. c) The requirements of the integration process, such as minimal distances
between diffusion edges, contact holes, etc, are not impediments to the design of optimal devices.
At this stage these items cannot be fully understood. Therefore, a closer inspection of the operating principles of CDD's is indispensable. To that end, Chapter III discusses various aspects of error compensation together with their practical consequences, while Chapter IV deals with practical design examples.
1-3 Some basic CDD-configurations for non-linear signal transfers
1-3-1 Introduction
For the design of CDD's for non-linear signal transfers it is necessary to have certain basic configurations at one's disposal. These have partly been found beforehand from configurations which basically perform linear
transfers. Further experiments with certain realizations of designs for
non—linear transfers have led to better insight and consequently to other
(new) basic configurations. A multitude of basic configurations are possible. Possible variables are, for example, the shape of the emitter or the collector(s) and buried layers, the number and the positions of the bias injection contacts, the way these contacts are driven, and so on. Further variables can be inserted by making the different layers inhomogeneous.
Only the most obvious basic configurations are dealt with in this work. In all but two of the cases, only one of the geometrical parameters of the transistor structure has been made variable. The upcoming sections deal in detail with the selected basic configurations.
1-3-2 CDD's with curved_base/emitter_geometr^^
Figure 1-6 depicts the basic configuration with its cross-section.
SP-diffusion
-collector-base area
possible emitter
bias injection contacts EPI-layer
DN-diffusion DP-diffusion
Figure 2-6. Basic configuration for non-linear transfers.
The non-variable parts are drawn in solid lines. The variable part is here the emitter path. One possible emitter path is indicated by dashes. The emitter is always curved symmetrically with respect to the x-axïs, The (low-frequency) transfer function does not change if the whole configuration is enlarged or reduced. A variant of this basic geometry is shown in Figure 1-7, in which the part below the x-axis has been deleted.
In comparing the configurations we observe that there exist some small differences in the neighborhood of the current injection contacts. As the configuration in Figure 1-6 can be regarded as two equal configurations, shown in Figure 1-7, which operate in parallel, there are no other essential differences between the two configurations.
Figure 1-7. The geometry of Figure 1-6 having been halved.
As will be discussed in Section 1II-4., a large series of non-linear transfer functions can be realized with these basic geometries. It is theoretically feasible to realize any arbitrary function, tempting one to draw the conclusion that other basic configurations would be superfluous, however, one drawback of the above basic configurations is that the realization of functions with relatively large or small slopes create practical problems. Therefore, other configurations will be taken into consideration too.
1-3-3 CDD^s_with_curved_base/emitter gcometr^i_symmetricallY positioned bias_inj_ection_contacts_and annular buried layer ■
The variable part of this class of CDD's corresponds to that of Figures 1-6 and 1-7. The difference with respect to the above-mentioned geometries concerns the geometry of the collector region and its contacts. The col lector region consists of an annular buried layer with point-like contacts on botli sides. Figure 1-8 shows the basic configuration with one possible (dashed) curved emitter.
The geometry may be halved here as well by virtue of the reasoning pre sented in discussing the previous configurations. With respect to the transfer functions the main advantage this configuration has over the previous model is that a larger value of the derivative can be realized at the beginning or at the end of the transfer function. An example will be given in Section IV-5.
Each function produced by one of the discussed basic configurations must belong to certain function classes (see Section III-3).
bias injecti contac
annular burled layer
collector contact 2 possible emitter base-collector layer
Figure 1-8. Basic configuration with annular buried lager.
1-3-4 Bas^_c CDD-con figurations with_asyjmmel_r icallv_gosi L ioned^bias infection contacts,
All CDD-configurations of Sections 1-3-2 and 1-3-3 have symmetrically positioned current inj set ion contacts, A fundamentally different configu ration appears as soon as these contacts cease to lie symmetrically. As a maximal sensitivity of the potential distribution on the base perimeter to the bias currents is desired, changes of the positions of the edge in
jection contacts are left out of consideration. Such changes would entail less accurate transfer functions. A deviation of the central injection contact, however, opens up new possibilities._In this way one can obtain a further degree of freedom. On the one hand, the flexibility, i.e. the number of transfer functions which can practically be realized, is in creased. On the other hand, the mathematical problems are considerably enhanced, since both variables (the variable shape of the emitter locus and the variable position of the central injection contact) will certainly be interdependent, so that two mutually coupled potential distribution problems will appear. In Section IV-5 a practical device in which this more complex basic configuration is applied will come up for discussion. Only a shift of the central injection contact along the x-axis is considered. Figures l~9a-b depict examples of these basic configurations.
Figure l-'Ja Figure 2-9b
Figure 1-9. Examples of basic configurations with variable emitter locus and variable central injection contact.
-15-CHAPTER II THE MODELING OF VARIOUS BASIC CDD-CONFIGURATIONS
II-1 General
In Section 1-2 we have introduced a new design approach for CDD' s. Two models were used, namely a simplified and a so-called mirroring model. Wi th the simplified model inaccurate designing was feasible, whereas the
errors could be analyzed with the mirroring model. Accurate designing
could be achieved by having both models interact in an iterative system.
In this chapter the basic configurations dealt with in Section 1-3 are supplied with their appropriate simplified and mirroring models. Further more, the mathematical methods appropriate to the simplified models are given. In Chapter III the numerical methods in connection with the mirror ing models are extensively treated.
II-2 The modeling of some basic configurations selected earlier The basic configurations mentioned in Section 1-3 have been selected on practical grounds (Figures 1-6 through 1-9). In the 1ight of mathematical considerations, however, a slightly different selection is more convenient. The possible configurations complying with Figure ]-6 can be regarded as a subset of all possible configurations of Figure 1-9a. The same holds for Figures 1-8 and l-9b. This contemplation results in two generalized mathe matical problems. In the interest of enhancing practical insight, apart from the treatment of these problems, two special geometries will be sub-jected to closer examination.
Figures 2-1 a and 2-lc depict two possible basic configurations. If the position of the central injection contact and the shape of the emitter locus are assumed to be arbitrary, they correspond to the same possibilities as the configurations of Figures 1-6 through 1-9. Figures 2-lb-d show two special subsets of the sets of geometries according to Figures 2-la-C. The emitters have an annular shape, whereas the positions of the central in jection contacts remain variable.
The question now arises of how to model these devices so that they are accessible for mathematical treatment as simple as possible. As a first
-16-step the basically three-dimensional structure is regarded as two-dimensional. Smith [2-1] thoroughly investigated the admissabi1ity of this important step and concluded Chat no noticeable errors were intro duced. As a consequence all models can be drawn in a two-dimensional plane and the potential problems which arise are two-dimensional.
Another simplification occurs if it is assumed that the bias currents do not slip underneath the emitter and if the bias-current injection contacts are considered as points. These simplificat ions cannot entirely be disregardec since they may give rise to non-negligible aberrations in the calculation of the transfer function. However, the emitter may be considered as a curved line acting as an absolute boundary for the base bias currents if the base diffusion is restricted to the area inside the
emitter locus. In that case the domain path may be assumed to coincide with the inner emitter edge. Likewise, the bias current contacts may be con sidered as infinitely small if their actual perimeters have some special shape (see Section II-4). If these precautions are taken no noticeable distortion in the transfer function will occur.
In spite of the advanced model simplifications resulting from the above approximations the thus obtained model is not yet generally accessible for those mathematical manipulations which are necessary for design purposes. Further simplifications lead to considerable deviations between the transfer function of the model and the actual device. Therefore, for the time being further simplifications will not be carried out and the resulting model will be considered as the mirroring model. The analytic methods for ana
lyzing it will now be as simple as possible. In order to find suitable "simplified models" further simplifications must be carried out until straightforward mathematical methods for design purposes can be found. These simplifications imply:
a) The base area is regarded as infinite. The presence of the emitter locus and the collector region in the structure and hence their effect on the potential distribution in the base layer are ignored.
b) The domain width is regarded as infinitely small, so that the emitter
current is assumed to enter the collector area in one discrete point. This point corresponds to the position of the potential maximum on the base perimeter.
c) The edge bias-current injection points are assumed to coincide with the emitter locus. (In practical devices this is impossible since this would imply a short-circuit between these injection points and the emitter. The minimal distances are dictated by the demands of the applied i.e. process.)
If al] model simplifications are introduced the resulting simplified and mirroring models take the form depicted in Figure 2-2 a through 2-2 d. The upper row of drawings depicts the basic configurations given earlier in Figures 2-1 a through 2-ld. The middle row represents their mirroring models and the lower row their simplified models.
II-3 Straightforward analysis of the simplified models
lir3-_^_General
If the models of Figures 2-2a through 2-2d are inspected, two two-dimen sional potential problems are encountered in all cases: one concerning the base layer with its bias-current inj ection points and the other con cerning the collector area or buried layer with their contacts.
Theoretically there exists a link between both problems. This is formed by the recombination current in the base of the conducting part, and the barrier currents in the isolating part of the transistor structure. In all cases, however, the effects of these currents can be neglected.
For this reason, both potential problems can be treated separately. As will be shown, the potential problem in the collector (buried)layers can rather simply be solved. The problems concerning the base configurations appear to be much complexer.
II_l3-2 §traightforvard_analy_sis_of_ kuried_lay_ers ÏH£lyrïing_th^ir_contacts
With reference to Figures 2-2a through 2-2d, two fundamentally different collector configurations can be distinguished, namely that of Figure 2-2a and 2-2b and that of Figure 2-2c and 2-2d. These two structures have been redrawn in Figure 2-3 and have been placed in a plane with cartesian and polar coordinates.
Figure 2-5 The current division in the collector area or buried layer
Assume now that at an arbitrary point P (r,i|0 in Figure 2-3a a current i =c e e a 1 is inj ected and the contacts c. and c, have been
\ I
circuited, (e.g. by the inputs of a balanced current follower) It has been shown by conformal mapping methods [2-]] that the differential collector current i = i , - i ,, meets:
r cos éi (2-1)
or, if the normalized transfer function of the CDD is substituted into (2-1) (see Section 1-1):
f (x) = r cos 'p (2-2)
It is striking that this differential current is not dependent on the y -coordinate of the point P(r,0j).
The buried-layer configuration of Figure 2-3b gives rise to another equation for the differential collector current. Once again, with a conformal mapping method [2-1] it can be shown that in this case
"c2 = (1
'li
-20-or like above:
Y = f(.x) =■ 1 - & (2-4)
hence, the differential collector current is now independent on the radial component of the injection point.
11-3-3 Mathematical consider a tions_of_th^base-emitter_geometries The base-emitter geometries in Figures 2-2a and 2-2c can be regarded as elements of one general set, having two variables: the shape of the emitter and the position of the central injection contact. The calcu lation of the emitter geometry implies the integration of a differential equation. As this has to be executed by numerical methods, the solution gives little insight into the correspondence between certain geometries and their transfer functions. Therefore two subsets will be considered (Figures 2-2b and 2-2d). The simpler cases appear to be easier to solve. In these cases a direct connection between the change of the remaining variable (the position of the central injection contact) and the re sulting transfer function can be found. We start with these simple models.
IIz3-4 Thejnodel_of_Figure_2-2b
With the aid of the simplified model of Figure 2-2b, which has a fixed circular emitter structure, a rectangular collector structure and a variable central bias injection point, a closed relation between the geometry and the (approximate) transfer function can be derived. If the distance between the central injection point and the origin of the co ordinate system is denoted as a_ the resulting transfer functions are
(Appendix A)
2 1 +5 -2
(X~b)(Y+b) = l-b , where b = ~y~~ >
and where Y is the (normalized) output variable (2-5)
The geometric representation of (2-5) is a set of (parts of) orthogonal hyperbolas depicted in Figure 2-4 for different values of a. Strictly speaking these hyperbolic -transfers are of little practical significance. As a starting point for the design of some other special transfer functions, where a second variable is introduced, they can, however, have some advantages (see Section IV-3).
Figure 2-4 The transfer functions belonging to the simplified model of Figure 2~Sb.
If a in equation (2-5) is chosen to be zero, the transtcr function degenerates to a straight line. In this special case a very interesting device is found: If the emitter current is used as a second information carrier, which can be achieved by modulating this current according to ig1^1) = X'l with (0 < X' C 1 ) , a two quadrant analog multiplier has
been realized, because the differential output current is proportional to the product of the variables X and X'. (In Chapter IV the independent variable X' will be denoted as Y and the output variable as 2, so that
'L = f(X,Y).) If two of these devices are driven with opposite signals,
the combination can operate as a four-quadrant analog multiplier. Design examples have been given by Gilbert [2-2] and Smith [2-3]. Both of these designs suffer from a relatively large amount of undesirable modulation products caused by a relatively large non-linearity in the transfer from X to the output signal. Section III-2 contains a new design method for such devices and Section IV-3 an example of a practical device with a greatly improved linearity.
II-3-5 The_model_of_Figure 2-2d
With this simplified model, in which an annular buried layer is used instead of a rectangular collector, the following set of (approximate) transfer functions for different values on a holds (see Appendix B ) :
Figure 2-5 The transfer functions belonging to the simplified model of Figure 2—2d.
The geometric representation for different values of a is given in Figure 2-5. If a is chosen to be zero the corresponding function is a part of a negative arc cosine function. Like the preceding functions, these
functions have little practical meaning by themselves.
The functions in Figure 2-4 as well as those in Figure 2-5 have one interesting property: the value of the first derivative can take values with broad limits. This is of great significance if power functions have to be realized (see Chapter IV).
ÏI~3-6_ Ï^Ê_m°dels_of^Figure_2-2a_and_2-2c
In these classes of CDD's the emitter locus ceases to be annular: its shape is arbitrary, although with the restriction that it must be sym metrical with respect to the X-axis. The central bias injection point may have the same asymmetry as in the preceding CDD-series,
In Figure 2-6 this base model has been placed into cartesian and cylindrical coordinates. As explained in Section II-2 domains are considered here as points corresponding to the potential maximum along the emitter locus. The problem now is how to find the link between the (approximate) domain loci and the approximate transfer functions.
infinite base area
Figure 2-6 Geometry for deriving Equation 2-7.
In Appendix C it is shown that all domain loci approximately obey the following first order differential equation:
A(r, a,, a) + X.B(r, *, a) 3r
H C(r, j , 3) S.D(r,
a)
(2-7)
A through D are the following functions of r, $ and a:
A(r, $, a) = - r s i n $ { a ( ] + r ) 2 a r cos t 2 r cos y ï ,
B(r, a) = r sin 'p{a + a r~ lar cosiji + 2 a r cos $ -C(r, q>, a) = a (r+r ) +
acos^>(-2
■ 2 a r cos $ ) - r - r + 2r COS $ D(r, $, a) cos 'p - 2 a r cos ~ $
i
$-Further specification of equation (2-7) is obtained if the dependence of the parameter X on r and $ is known. This is found by using Equations (2-2) or (2-4), found earlier when considering the collector configuration. Let's say, for example, that the present base-emitter configuration is combined with the collector configuration given earlier in Figure 2-3a, complying with the function f(X) = r cos 4' (Equation 2-2), where f (X) is the desired transfer function of the CDD. Now the combination of Equations
(2-7) and (2-2) can easily be solved with the aid of common methods of numerical analysis. The solution gives the inner edge of the emitter locus complying with the approximate transfer function. Examples will be given in Chapter IV.
A special class of CDD-models is found if Che central bias injection point is placed symmetrically between the edge points (see Figure 1-6), i.e. a = 0. In that case Equation (2-7) degenerates to the following equat ion.
, 3
sin 4' cos
X(r cosy " r cos ■[.)
(2-8)
Again, this equation can be combined with the equation that holds for the chosen collector configuration. Assume we have a collector structure for which Equation (2-2) holds and we wish to design a CDD with an approximate transfer function f(X) = XJ; then for X in Equation (2-8) we have to substitute : X ■= r COS 4 , i.e. the inverse function of f (X). The
resulting differential equation which has to be solved numerically then yields:
3 2 2 2 4 .
j_r_ _ -2 r sin $ cos $ + r cos tf(r sin $ + r sin?) (2-9)
a* , 2 ~T~^ TT 2 T77 ~T~ 3 ~ 7 7 - l - r + 2 r cos $ + r cos ?(r cos 9 - r cos 9)
If we have a collector structure for which Equation (2-4) holds (annular buried layer), the ultimate differential equation which holds for one special transfer function can be derived by substituting, Equation (2-4) into Equation (2-7). The remainder of the procedure equals precisely that used with the previous model.
With special configurations within the classes depicted in Figures 2-2a through 2-2d, using Equations (2-7), (2-2) and (2-4), any arbitrary non linear transfer function belonging to certain classes can theoretically be realized. These function classes will be further specified in Section 111-3. If practical devices are designed according to these methods, however, we will always observe considerable differences between the pre determined function and the ultimately realized funet ion. The reason for tli is is that t!i e models have been simpl ified too much. In order to get
Figure 2-7a.
Figure 2-7b.
Figure 2-7. Two early practical devices which have been designed with their simplified models.
some impression of the nature of these deviations we will first discuss a few early experiments, gradually working up to the more sophisticated design methods, which will be dealt with in Chapter III.
Figure 2-7 shows two examples of such devices intended for non-1inear signal transfers. Both devices are operating with the maximal admis-sible average bias current (1 - 10mA) in order to assure domains as narrow as poss ible. In this way the models can best be approximated by the devices. Figure 2-7a shows a CDD intended for the signal transfer f(XJ = X . Its model is one of the possibilities included in the con figuration of Figure 2-2a. Figure 2-7b shows another configuration whose model is given in Figure 2-2b, It produces approximations of some hyperbolic transfers.
Figures 2-8 and 2-9 compare the measured transfer functions with the desired ones of the devices depicted in Figures 2-7a and 2-7b,
Y°Ki/A Y
-M
CENTRAL INJECTIONCON'ACT:
DESIRED FUNCTIONS
Figure 2-8 Measured transfer functions of the device shown in Figure 2—7a.
Figure 2-9 Measured transfer functions of the device shown
in Figure 2-7b.
In order to get some impression of the dependence of the transfer on the average bias current I , and hence the domain width, the transfers
in Figure 2-8 are given for different values of I , namely 2.5, 5 , 10 and 15 mA. As expected, the approximation of the desired transfer function improves with increasing bias current I . If this current is chosen too large, however, the transfer is distorted by secondary breakdown of the base-emitter junction, This occurs in Figure 2-8 at bias currents larger than 10mA, just in that part of the base-emitter path where the base-emitter voltage V is maximally negative. The domain is then positioned near one of the edge inj ect ion contacts. As a consequence, the maximum negative base-emitter voltage occurs just near the other edge inj ection contact. The risk of secondary breakdown can be coped with by giving the injection contacts a special shape. This can be considered as a first step toward improving the quality of the devices and is the main topic of the next section.
Lastly, from Figure 2-9, we observe, indeed, that the attainable maximum (minimum) values of the first derivative are pretty large (small).
We will now discuss the accuracy of the functions produced. We observe
from Figures 2-8 and 2-9 that rather rough approximations of the desired transfer functions are produced. With the more advanced design methods which will be described in Chapter III it is possible to compensate the distortion. In addition, the design methods allow accurate designing with
specific (rather low) injection currents. Hence, the power consumption of the resulting devices is considerably lower than that of the devices described in this section.
II-4 Optimal shape of the bias injection contacts
There are certain practical limitations concerning the choice of the absolute and relative dimensions of the bias inj ection contacts. The devices match their mirroring models the best if the bias injection contacts are infinitely small. However, this is not possible on obvious practical grounds. If the contacts were really pointed the local poten tials in the base-layer would assume infinite values. Besides, the gradient of the potential, i.e. the electric field strength, would assume infinite values as well. The absolute value of the potential on the perimeter of the base layer is always maximal at an edge inj ectIon
contact if the domain is situated near the opposite edge injection contact (see Figure 1-3). As the emitter lies very close to these contacts, the maximal value of the reverse base-emitter voltage V will first be exceeded at those spots if the average bias
BE max
current I increases. The previous section's Figure 2-8 shows a transfer function of a CDD where the base-emitter breakdown voltage has been ex ceeded. Besides, if the electric field strength in the base-layer ex ceeds a certain value, the majority carriers inside the base layer will be subjected to mobility modulation. This condition violates the linear resistive nature of the extrinsic base. The most adverse situation occurs here at the perimeter of the central injection contact.
Smith [2-1], [2-3] has shown that the practical potential field in the base layer does not noticeably deviate from the theoretical situation, While the maximum base-emitter voltage is prevented from being exceeded, when the contact radii exceed certain minimum values and when the boundaries of the contacts coincide with equipotential lines.
A further important property of practical CDD's, involving the dimensions of the injection contacts, is the power consumption in the base layer. For adequate domain operation the average base bias current 1 must be large. As a consequence, an overwhelming part of the CDD-chip's total power consumption is taken up by the dissipation in the base layer. The poten tial gradient is maximum near the bias injection contacts. Hence, most of the power dissipation occurs in the neighborhood of these contacts. So "hot spots" exist at those places. As has been shown in Section 1-1 , the domain width, which is of major concern for the transfer function, is in versely proportional to ( p i ) . On the other hand, the power dissipation in the base layer is proportional to I p .
Therefore, with regard to the power consumption of the chip, it is advan tageous to choose large values of p and small values of I . Unfortunately,
only two values of p arc available in a bipolar standard process, namely those of PisP and NPK devices. Furthermore, the power consumption can be restricted by choosing relatively large injection contacts. But a negative effect of choosing the relative dimensions of the edge base
contacts too large can be that some distortion of the transfer function occurs when the domain approaches the limits of its range.
These limits are positioned close to the edge base contacts. As will be explained in Chapter IV, PMP devices are only interesting when CDD's are used as comparator devices with relatively low bandwidths. Therefore, the demands made upon the dimensions of the injection contacts also depend on the chosen application of the devices. Because multiple comparators are realizable the best when the domain is as small as possible, the power consumption plays a more important role here than in NPN devices intended for accurate non-linear transfers, where distortion above all must be avoided.
If all. the above-mentioned aspects concerning the dimensions and shape of the injection contacts are accounted for and the sheet resistance and average bias currents have been chosen, the optimal normalised radii of
Figure 2-1Oa Figure 2-1 Ob
-30-the base contacts can be calculated [2-3]. Good compromises are reached when the contact radii arc chosen such as in Figure 2-10a for an NPN device in a bipolar standard process with p = 200 &/square and as in Figure 2-10b for a PNP device in the same process (p - 150Ofi/square). In these examples the base layer is circular with a radius of 125 um. Measurements have shown that these dimensions can be maintained in all other geometries occurring in this work.
-31-CHAPTER III ITERATIVE ANALYTIC SYSTEMS FOR DESIGNING CDD'S WITH ACCURATE NON-LINEAR TRANSFER FUNCTIONS
XII—1 Introduction
As discussed In Section 1-2-2 two types of models must be treated if a CDD has to be designed, namely an accurate (mirroring) model which is not accessible for direct designing and a simplified model which has no such restrict ion but is prone to relatively large errors. The simplified model can be used for designing the approximate geometry and the mirroring model can be used for analyzing the errors , thus producing the necessary information for subsequent iterative adaptation. The iterative design process, using these two models together, makes it possible to design accurate CDD's. This iterat ive method comes down to adapting the simplified model until the remaining errors are small enough to be acceptable. This chapter deals with a CAD-method which can carry out this task. The dis cussions in this section through Section III-3 concern only basic configurations with one variable geometrical aspect. In Sections III-4 and III-5 basic configurations with two variables are dealt with. Lastly, Sect ion II1-6 discusses the analysis of the mirroring models. Section II1-7 has a somewhat individual character: it discusses the calculation of planar resistors with the aid of one of the foregoing analytical methods.
III-2 The design of CDD's with variable emitter locus and symmetrically positioned central inj ection contact
We start by presenting a simplified block diagram which illustrates the basic operations. Next, we give a more detailed description of the design method.
Block A in Figure 3-1 represents the desired transfer function. The ultimate design of the CDD is represented by block F. The actual iterative process takes place in blocks B, C, D and E. In block B the straightforward calculation of the approximate geometry with the aid of the simplified model is carried out, whereas the error analysis, with the mirroring model is carried out in block D with the aid of iterative analytic methods. Lastly, block F calculates the necessary adaptations to the transfer function.
Like in any correct design method one has to start by defining the design obj ective. In this case the obj ective concerns the realization of the
0t h iteration desired transfer function adaptation transfer function "TV-designing with simplified model approximate geometry realized desired transfer needs adaptation
D
determination transfer with mirroring modelFigure 3-1 Bloakdiagram of the iterative design system (CDD's with one variable).
desired transfer function. Let's say this function is given by Y = f(X) Next, a suitable couple of simplified and mirroring models are chosen. The following step is a calculation of the approximate geometry of the emitter locus with the differential equation given by (2-8) together with the equation which holds for the collector geometry (see Section
II-3). The inverse function X * f (Y) has to be substituted in these equations. The total calculation procedure is represented by block B in the block diagram. Now the real transfer function is numerically calcu lated with the aid of the mirroring model (block D ) . With reference to Appendix D this function can be calculated by one of the following formulas: f(X ) JD exp- /dr.2 ,i , ,, {-—) J r cos i dtp dip (3-1) f(X ) V-(r, ♦, X ) J exp - C^)2)'(l dip (3-2) B o . 2 ^dr./, ,, xp ^ [r ♦ (^, I ii
Equation (3-1) holds for models with rectangular collector structures (see Figure 2-2a) and Equation (3-2) for a collector which has been supplied with an annular buried layer (see Figure 2-2c).
Mostly Lite results of these calculations will reveal differences between the originally desired funct ion and the tune t ion actually produced. The next step is inverting the errors. So the originally desired function f(X) is multiplied by a correction function ti(X) . In the next iteration the new (adapted) "desired" func tion is defined as
f'(X} = f (X) h(X). (3-4)
Now the inverse function of (3-4), X = g(r,y), must be substituted in the differential equation given by (2-8). Next the realized transfer function is calculated once more with (3-1) or (3-2), after which the errors will generally be smaller. This procedure is repeated until the errors converge to certain final values. Then the final geometry of the emitter locus has been realized (block F ) .
The main bot tleneck of the above method is the execution of the calculations (3-1) or (3-2). The reason is that the potential values V (r, $, X ) have to be known. These are the potentials at the perimeter of the base plane at a certain value of the input variable X = X . Various methods for obtaining these potentials are dealt with in Section 111-6.
I1I-3 Restrictions which have to be imposed on the functions to be realized
Though the above-described method is highly universal, there ex is t a few restrictions with respect to the transfer functions to be realized. The differential equation (2-7) holds for any arbitrary curve on the normalized base-col lector structures of Figures 1-6 and 1-8. Hence, the relation X = g ( r , ip), which has to be substituted in this equation (Section 111-2) before the iterative procedure can be started is, theoretically speaking,
allowed to be an arbitrary non-linear function which is the inverse function of the desired one. Clearly f(X) must be a function whose inverse is defined on (a part of) the interval [-1, +1]. However, the function should comply with a few other criteria, some of which imply fundamental restrictions. Obviously, the function must be normalized in such a way that it fits into the models. This requires the independent variable X to span the interval [-1, +1]- At the limits of this range the domain centroid is driven into the utter pos it ions of the emit ter locus, as close as possible to the edge injection contacts. Therefore, the Y-values corresponding to the extreme values of X must have values of ±1 as well. In addition, the function must meet the following demands:
i) The function must be continuous (not necessarily differentiable) and continuously rising.
ii) The function must meet the condition f(X) = -f(-X) (odd function). This condition can be omitted if two geometry-aspects are variable
(see Section III-4).
iii) The first derivative must lie between certain minimal and maximal values.
The last condition can be explained as follows. The domain will never be contracted into one point unless infinite base bias currents were to flow, hence, the domain will be spread up around the potential maximum. In practice, this means that the response of the collector currents to variations of X cannot be infinitely large, which implies that the first derivative of the function is finite. What its actual limiting values are depends highly on the basic geometry which is chosen. In the first series of GDI)' s which were designed according to the iterative design method described above, the basic configuration of Figure 1-6 (Section 1-3) was used.
In so far as the functions to be realized have only gently rising slopes, this basic geometry is pre-eminently suitable for the realization of accurate non-linear transfer functions (see Section IV-A). The maximum and minimum values of the first derivative that can be realized have been
investigated by two separate practical experiments with the given basic configuration, The experiments have been carried out with a (rather low) average bias current (Ip = 5 m A ) , Larger values of I result in somewhat larger attainable values of the first derivative.
The device of the first experiment was designed for a desired transfer function Y = X! . After several iterations the resulting transfer converged to a final curve. This curve is shown in Figure 3-2 together with the desired function. Apparently, the chosen model configuration does not allow values of the first derivative larger than 2. A similar experiment was carried out to find the lowest possible value of the first derivative with the given basic configuration. The latter device was intended for a trans ■
2
ter Y ■ X . After having obtained convergence in the iterative procedure the minimal value of the first derivative appeared to be | (Figure 3-3).
^realized transfer
Figure 3-2 Pra.oti.cal determiriation of the largest attainable initial value of the first derivative.
realized transfer des TBd transfer minimum slope ".5
Figure 3-3 Practical determiriation of the smallest attainable initial value of the first derivative.
*)
Both functions are only considered m the first quadrant: the total (odd) functions are f(X) * sgnjX] and sgnjXI .
Figure 3-4 Limits between which the initial slope of the transfer must lie.
By combining the above results, we can conclude that the transfer functions to be realized must have an initial slope which lies within the shaded area of Figure 3-4. As will be discussed later, this slope can be enlarged by choosing another basic configuration.
III-4 The design of CDD's with variable emitter locus and asymmetrically positioned central injection contact
A new class of CDD' s in which a true design of the geometric structure
belonging to a previously desired transfer function is possible, arises if both the emitter locus and Che position of the central injection contact are made variable. The concerning models were earlier given in Figures 2-2a and 2-2c. Due to the fact that two aspects of the geometry are independent ly variable, the calculation procedure is much complexer in this case. This leads to a more complicated iterative procedure, which has schematically been shown in Figure 3-5. Now the iterative procedure contains two loops, one for the determination of the emitter locus and one for the corresponding position of the central inj ection contact. In block A the desired transfer function is again chosen. Block B calculates the approximate emitter locus with the aid of the differential equation (2-7) in Section II-3-6, However, the solution of this equation now depends on the chosen start position of the central injection contact. A wrong choice of this position leads to an emitter locus that start at $ a 0 i C = I a nd ends at $ = 180 , r f I.
Figure 3-6 illustrates this phenomenon, where three solutions of the differential equation corresponding to three different positions of the
0th iteration
check of transfer with mirroring
model
Figure 3-5 Extended iterative procedure for models with two variables.
-0.4 (too for from origin)
orroct position) 3 [too close to
Figure 3-6 The effect of wrongly choosing the position of the central injection contact.
-38-central injection contact are given. Therefore, Lhe second lucrative loop is introduced (.blocks II and G) . As soon as it is noCed than the emitter locus does not arrive at r = I, the central injection contact is shifted by a certain amount along the x-axis and the differential equation is solved once again. This is repeated until the correct position of the central contact is found and the locus finishes at r = I. So, every "large" iteration (blocks B, C, H, D and E) contains a number of "small" iterations (blocks C, H, G and B ) . It will be clear that this result in an enormous increase in required computing time. The question might arise of what the practical consequence is if the emitter locus does not arrive at r = I at $ = ISO . The answer is that either the entire interval of the function is not realized (r > 1 at ■;■ = 180 J or the CDD is not realizable at all (r ': 1 at ■; = ISO ) , because in that case the left injection contact would lie outside the emitter locus.
II1-5 The application of the extended iterative system with two different collector geometries
In section II-2 two different mirroring models with variable central inj ection contacts have been introduced. The difference between the two models concerned the collector geometry. One model contained a rectangular collector and the other an annular buried layer. As we have started with the assumption that the problems concerning the base-emitter geometry can be regarded as independently of the problems concerning the col lector geometries, the considerations of the previous section hold without re strictions for both collector geometries.
III-b The analysis concerning the mirroring models
III—6—J Introduction
As was extensively described in Section III-2 the most essential step in the iterative procedures is checking the transfer function produced with the aid of the mirroring model from the domain locus which was calculated with the simplified model. This step is the shaded block in the simplified block diagram of the procedure as it is redrawn in Figure 3-7.
Figure '6-7 Simplified block diagram of the iterative system.
Figure '6-Ha Figure S-8b
.Vn?
Figure 3-8 Two typical examples of the potential problem that occurs in the iterative procedure.
-40-The most cumbersome part of this step is the calculation of the voltages along the base perimeter, given the necessary bias currents flowing into the base. This reduced problem, two typical examples of which are shown in Figures 3-8a and 3-8b, can be solved in various ways. Which method is ultimately chosen depends on the nature of the geometry and the desired accuracy. It is sometimes possible to carry out the task by closed mathematical methods, if this is not feasible, which is frequent ly the case with the geometries dealt with in this thesis, empirical or numerical methods are called for. The upcoming sections describe most of
the possible systems and discuss their characteristics, advantages and disadvantages.
111-6-2_ Modeling_with_conduc£ive_ria2er
Several decades ago, the so-called "potential trough" was sometimes used as a model, to solve various field problems. A two-dimensional variant of this is a sheet of conductive paper. Basically, this can serve excellently as an electrical simulation of the base plane and its contacts. As the electrical behavior is invariant with respect to linear enlargement and because high-conductivity spots (the simulated contacts) can be made by silver paint it is possible to make very large models of the base plane which are accessible for spot measurements of the potential on the base perimeter. The main drawback is, however, that, to the best of the author's knowledge, high-precision conductive papers with very low inhomogenei ty and anisotropy are not available.
Still, in the absence of better methods, our first CDD' s have been designed with the aid of models of conductive paper L3-1J. The inaccuracy of the method, however, appeared to be in the order of I to 2%, due to the inhomogeneity of the papers.
Til—b—3 Q2HË°EÏÊi:_rD^EEinË
Using conformal mapping methods as a tool for solving the potential problem in the base layer implies that a two-dimensional geometric representation of the base layer, with its inj ection currents, is mapped onto a complex plane which is commonly called the z-plane. General1y speaking, one attempts to find a complex transformation that is able to
4 !
-map the original geometry onto another complex plane (the w-plane) so that tli e geometry is s impl if ied in such a way that a direct cal culation of the potentials in the w-plane is possible. The potential formula found in tli is way is then transformed by means of the inverse transformation formula back to the z-plane. The real part of the potentia f o rmula obtained in this way gives the exact values of the potentials in the z-plane. Although a correct transformation formula does always exist, the main difficulty consists of finding it. Direct methods for finding it do not exist [3-2 J. Therefore, the method is suitable for a restricted number of simple geometries only. Gilbert used a conformal mapping method to analyze a (semijcircular base plane [3-3 J. In more complex configu rations , however, the transformation formula has to be found by numerical methods [3-2]. These methods, however, have fallen into disuse upon the
rise of other numerica1 methods such as elements methods, finite-difference methods and the use of integral equations [3-41, [3-5], [3-6].
III-6-4 Finite-difference and finite-elements methods
Finite-difference and finite-elements methods have grown very popular in recent years for the following reasons:
a) Any field problem that is described by Laplace's or Poisson's equation can basically be solved with these methods.
b) The methods are extremely suitable for calculations performed with the
aid of digital computers.
The two methods bear certain resemblances with respect to their starting points and the ultimate computer calculations, the theoretical background however, differ. In the course of this work, the finite-difference method in particular has been examined with respect to its applicability to potential problems in CDD-configurations. This study has resulted in some more or less practicable techniques. For a good understanding of these techniques some marginal theoretical notes are inevitable. However, we will restrict the analytic considerations to those that have a direct bearing on our practical aims.
-42-II1-6-5 Calculating_the E°^.^BÊi§l5_Êl2nê_Ël]Ë_^£^0ËËIX-2f_a CDD base-disk with the_finite-dif f erence method
The problem Is first somewhat generalized. Figure 3-9 depicts an arbitrary base configuration as it might occur In aome CDD.
c
3Figure 3-9 Some CDD-base geometry for illustrating the finite-difference method.
A two-dimensional resistive layer with a sheet resistance of p ohms per square contains three highly conducting contacts: C , C? and C . The left and right injection contacts touch the boundary and the geometry is symmetrical with respect to the x- and y-axes■ These restrictions have only been made for the sake of simplicity: the method is suitable for any arbitrary geometry. First, the relevant area is divided into meshes. All meshes are congruent. Although cells with different shapes are possible, we have opted for square ones, the reason being that the square mesh is the most popular one and almost all sophisticated error analysis concern this type of mesh [3-5]. It is not necessary for the boundaries of the
-4
3-boundary meshes to coincide exac tly with the 3-boundary of the plane.
Interpolation formulas for non-fitting meshes on the boundaries are available. For the initial trial, however, the geometry was simply partitioned into meshes in such a way that the boundary of the base plane fits the boundary of a mesh everywhere, in other words , the length and width have been approximated everywhere by the nearest whole number of meshes. An initial simplif ication of the model, by which the necessary number of meshes is divided by two, is effected if only the upper part (the part above the x-axis) is considered. Due to the symmetry of the geometry with respect to the x-axis all potentials are doubled through this manipulation. In this way calculation time is reduced without loss of information.
original boundary
•'left hand injection contact
Figure 3-10 Detail of the upper part of the geometry in Figure 3-9 after being partitioned into meshes.
A further measure for making the problem accessible for calculation implies that the current sources are connected one by one and that the calculation is carried out twice, after which the calculated potentials in any node are summed,
Although the injection contacts in the model of Figure 3-9 are fed by current sources, calculations carried out by the finite difference method are the most convenient if voltage sources are connected at the contacts
iB1-(i+x) iB iBB-Tï-x)i
Figure 5-11 The resistive layer replaced by three discrete resistors.
so that the initial potentials in the nodes of the injection contacts are known. Due to the 1inearity of the resistive system the only dif ference is that some scaling factor based on Ohm's law is necessary. This is elucidated by Figure 3-1 I. The (low—frequency) resistivity of the CDB~base, as it is seen from the inj ection contacts, is unaltered if the two-dimensional resistive structure is replaced by three resistors (R through R_). Before the first calculation is carried out the right-hand current source is deleted. Now the left-right-hand current source can be exchanged with a voltage source of
So the potential distribution is not affected if a suitable voltage source is connected instead of a current source. The values of R through R~, however, are generally unknown. This problem is solved by calculating the current that flows in the left-hand inj ection contact. As an example, the calculation procedure for the value of R„(= R-) is discussed. (A similar procedure holds for the calculation of R . ) . The procedure is illustrated by Figure 3-12.
As soon as the potentials in all the nodes resulting from a voltage V, with a fixed value (say 1 volt) have been calculated, the current flowing in the left-hand contact can be calculated. This is effected by drawing an arbitrary contour C that fully encloses the left-hand contact. Then the current I., according to Ohm's law, yields
— n .VV ds (3-5)
-45-Figure '6-12 The configuration ready for partition into meshes and calculation of RCl.
where p is the sheet resistance of the layer and n the unity normal vector. If the discretizing of the layer is taken into account, this equation changes into
I, = - S C V V - Vn), (3-6) 1 p n = | r+fir r
where N is the number of meshes along the contour C. As the contour is allowed to have any shape and the geometry is divided into square meshes, the contour can profitably be chosen to be part of a rectangle ( dashed
vl curve in Figure 3-12). The value of R9 is now found from R,- = ■=—.
I Calculation procedure
A possible geometry whose potential field has to be calculated is shown in Figure 3-12 (This example implies a geometry occurring in a design of a Cub for triangle-sine conversion). The shaded parts (A' through C') are highly conducting injection contacts. Between the contacts A' and B' a voltage of 1 volt is applied. The boundaries have been approximated by rows of whole numbers of meshes (see Figure 3-10). The calculations aim at finding out the potentials in any node and are carried out by the Successive Over-relaxation Method (SOR), which has been extensively described in literature [3-5]. The method operates as follows. In any node an approximation of the two-dimensional Laplacian equation
yx dy
