A New Concept in Calculation of Thick Film Resistors

Because of interaction of resistor and conductor material and variation of film thickness due to different resistor geometries, square resistance becomes a function of resistor length and width. Additional to the square resistance one paste parameter or two parameters must be introduced if the influence of both length and width on the square resistance is to be taken into account. From resistance measurements of a test pattern all paste parameters can be calculated as numerical functions of resistor length by a computer programme using the method of least squares. Beside the layout of thick film resistors, the data are suitable to give a computer plot of square resistance in relation to length and width. This representation can play an important part in paste evaluation and process control. It shows at a glance what relationship will be present with various combinations of conductor and resistor materials or by changed process conditions.


INTRODUCTION
The layout of thickfilm resistors of small geometries would involve considerable systematic errors if the square resistance only is taken into account as a fixed paste parameter. Because of the dominant influence of resistor length, most studies have not treated the influence of resistor width on square resistance) -6 In the cases in which this effect has been considered,  no clear relationship which would be of use in designing a layout has been established.
In the method described here, it is shown that in addition to the square resistance, one paste parameter (No. approximation) or two parameters (No. 2 approximation) must be introduced, if the influence of resistor width on the square resistance is to be taken into account.
All paste parameters can be rapidly determined, with the aid of a computer programme, as numerical functions of resistor.length from resistance measurements on a test pattern. It is important to emphasize that the terms length (L) and width (I40 of a resistor always mean the layout values, which can be taken into account only. (The dimensions on the substrate may be different to a certain extent, because of paste rheology etc., Figure 1).
The interaction of resistance material with the conductor connections will in most cases lower, but in some cases increase the square resistance of short resistors. In addition, the square resistance is 163 resistor geometry. The film thickness relations can be strongly influenced by rheological properties of resistor paste and by different screen printing parameters. Usually, however, short resistors show an increase in thickness, due to defective depression of the screen, while at the transition from a narrow to a broad resistor the film thickness likewise increases or forms a maximum.
Figure shows measurements of the film thickness as a function of resistor geometry. its cross-section were rectangular and its width equal to the layout value W (Figure 1). To separate the variables L and W, we use a product formula: teff(L, W) t elf(L, 1)" f(W) or by virtue of R 3 In order to determine the paste parameters as numerical functions of L, a test pattern containing 36 resistors is used (Figure 2). Each resistor length L 0.5, 1.0, 1.5, 2.5, 4.0 and 7.0 mm, is present in widths W 0.5, 0.7, 1.0, 1.5, 2.0 and 4.0 mm. To obtain statistical confidence limits for the values, 25 substrates with the same test pattern are used in each investigation. For the data, reported here, the conductor material is PdAg 9473 (Du Pont), pre-fired, thickness 13/am. The resistor material is series 1300 (Du Pont), fired at a peak temperature of 860C for 9 min to 10 min.

THE NEW RESISTANCE FORMULAS
Because of the interaction of conductor material and resistor material and because of the dependence of film thickness on geometry, the square resistance and the effective thickness teff are functions of the length and width of the resistor. The value of telf is the thickness which the actual resistor would have if The first term in Eq. (1.1), a function of length only, is termed square resistance of the unit width. It is the first paste parameter and epresents the curve of square resistance as a function of length but is now obtained for a definite width, namely the unit width mm. The unknown dimension less function f(W), for which a formula must be found, describes the dependence of effective thickness in relation to width for a fixed length, according to Eq. (1). The first intention for foe) was to find a function which will describe the relationship between thickness and width in general, like Miller has found i.e. an increasing curve which may have a maximum ( Figure 3)ff Beside that, the function f(W) must possess the following properties: 1) The function f(W) must contain unknown parameters the new paste parameters which determine the form off(W) for constant L so, that Eqs. (1) and (1.1) are satisfied. As the parameters are calculated for a fixed length from the 6 resistance means of the resistors with different widths, t elf in Eq. (1) represents the "electrically active thickness" as a function of W. This is a condition, transferred from Eq. (1) to Eq. (1.1).
2) The function f(W) must have a form making possible the use of a suitable mathematical procedure for calculating the unknown parameters at constant L.
Here f(I) is chosen in such a way, that the method of least squares can be used.
3) By virtue of Eqs. (1)   for three different degrees of approximation. rn 1, a 0 (zero approximation) and hence f(W) 1. This is the trivial case: film thickness and square resistance are independent of W, by virtue of Eqs. (1) and (1.1). rn 1, a 0 (first approximation) f(W) is a root function if rn > (e.g. in Figure 3, rn 1.2). rn 1, a -0 (second approximation). The function f(W) is now more flexible than in the first approximation and can describe a maximum in teff (W).
If rn < 1, the function f(W) can also describe the case in which the effective film thickness telf falls off with increasing width W. The reciprocal relationship gives the width dependence of square resistance. Its actual value is obtained when it is multiplied by the square resistance of the unit width mm, where the curves are always equal to one. The flexibility of the function f(W), which is able to describe a increasing, a decreasing or a constant curve of RE] (L, W) as a function of width W, according to the measured resistance mean values, is a essential of the method. In Table I,  The method has also been tried out with other paste systems-thick films, resinates and carbon pastes giving more or less the same distributions. Because method of least squares is used the mean error is equal to zero. Conditions in an actual circuit layout will unfortunately be different. The position of the resistors on the substrata may have some influence, the unevenness of the substrata surface for example caused by previously dried resistor prints causes some departure from the thickness relations. But only if a cross-over or a thick film capacitor is very close to a resistor area an additional correction is considered necessary. A squeegee movement perpenticular to that used here let decrease square resistance particularly for long resistors. For lengths from 4 to 7 mm the decrease in square resistance will  be about 5 to 10 percent. If this influence shall be taken into account another set of test circuits must be produced and evaluated.

DIAGRAMS OF SQUARE RESISTANCE
VALUES R m (L, W) If the computer system used includes an x/y plotter or a CRT display, the square resistance can be displayed as a function of length L for any required width W immediately after computation of the paste parameters R rn (L, 1), m(L) and a(L). The equations required are obtained by substituting the equations (4) or (5)  can be found in long resistors. The drop in resistance for short resistors is over-compensated by an opposite effect; microscopic examination of the resistance ffdms showed micro-cracks particularly in the neighbourhood of the conductor contacts. This increases the apparent square resistance, particularly for resistors with low L/W ratio. Paste 1331 in Figure 6 shows a clear influence of diffusion on the form of the curve, square resistance dropping for long resistors also. This behaviour is qualitatively the same for all pastes of the Du Pont 1300 series from R 1002/r to R D 100 k2[t, but high-ohmic pastes show a greater slope. The behaviour of the M2/m-paste 1361 (Figure 7) is surprising; the square resistance does not fall off for broad resistors, but it does suddenly drop for lengths less than 1.5 mm. Obviously this paste is not merely a version of paste 1351 containing more glass, a conclusion which is also confirmed by examination of the surface structure. The increase of square resistance which is always observed in narrow resistors in these pastes may be caused by the decrease in film thickness. We have however found complicated relationships between width and square resistance in some cases, especially with high-ohmic pastes. Although the square resistance of narrow resistors was mostly higher than that of broad resistors, the quantitative difference may be small. In some cases, even the opposite effect, a drop in square resistance with narrow resistors, can be observed. This is particularly the case when high-ohmic pastes are fired at high temperatures. The reasons for this phenomenon are still obscure.
The strength of the effect of screen printing conditions on the form of the curves of/(L, W) is shown in Figure 8, the condition involved here being mask thickness. The prints were made with a directly-coated screen and mask thickness of 0-3, 15 and 30/am above the screen. The saw-tooth structure of the edges, indicated here for mask thickness zero (which is often observed in screen printing), was obviously the cause of the sharp rise in R for narrow resistors. The larger drop in square resistance for short resistors can be contributed to the larger increase of paste thickness in short resistors compared to that in long resistors (with absence of mask thickness) and the fact, that silver diffusion occurs in the hole cross-section of the resistor, if its thickness is low. All three groups in Figure 8 were fired together and have the same palladium-silver contacts.