INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 37 (2004) 2170–2174

PII: S0022-3727(04)78411-9

A random resistor network model of voltage trimming C Grimaldi1,3 , T Maeder1,2 , P Ryser1 and S Str¨assler1,2 1

Laboratoire de Production Microtechnique, Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland 2 Sensile Technologies SA, PSE, CH-1015 Lausanne, Switzerland E-mail: [email protected]

Received 29 March 2004 Published 14 July 2004 Online at stacks.iop.org/JPhysD/37/2170 doi:10.1088/0022-3727/37/15/019

Abstract In industrial applications, the controlled adjustment (trimming) of resistive elements via the application of high voltage pulses is a promising technique, with several advantages with respect to more classical approaches such as the laser cutting method. The microscopic processes governing the response to high voltage pulses depend on the nature of the resistor and on the interaction with the local environment. Here we provide a theoretical statistical description of voltage discharge effects on disordered composites by considering random resistor network models with different properties and processes due to the voltage discharge. We compare standard percolation results with biased percolation effects and provide a tentative explanation of the different scenarios observed during trimming processes.

1. Introduction In the production of reliable electronic devices a very important aspect is the possibility of tuning the resistive components in order to adjust their resistances to a desired value within a given precision. Resistor values in fact usually deviate from their design values because of variations in the fabrication conditions, leading to resistance deviations as large as 10–20% for polysilicon resistors [1] or 20–30% for thick-film resistors [2]. The laser trimming method has been a widely used technique for calibrating resistor values. However, damage due to propagation of micro-cracks and the difficulty of the laser trimming of resistors buried in protective glass materials lower substantially the efficiency of this method. An important innovation in trimming technology has been the pulse voltage or current trimming method, in which a resistor is subjected to high electric fields capable of changing its original value. The high voltage or current trimming technique has been shown to overcome many of the deficiencies of the laser trimming method, leading to highly precise adjustment of the final resistance distribution, to less than 1%, for both polysilicon [3,4], and RuO2 -based thick-film resistors [2]. The phenomenology of the voltage discharge 3

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trimming technique is quite a rich one. Reversibility has been demonstrated in previously trimmed polysilicon films that can recover their original resistance under suitable tuned voltage pulses [3, 5]. Depending on the composition, the resistance of RuO2 -based thick-film resistors can be enhanced or lowered under identical voltage discharge profiles [6]. Resistance noise and aging can be seriously altered [7] and electrical failure can be driven to sufficiently high voltage pulses. The physical mechanisms leading to the voltage induced change of resistance depend on the nature of the resistors, and several models have been proposed. For example, in polysilicon films the resistance reduction upon voltage trimming has been ascribed to local structural modifications occurring in the grain boundaries due to the severe joule heating [8]. Local heating, which is thought to play an important role, also takes place within the glassy films [9] separating two neighbouring metallic grains in thickfilm resistors [10], and in the polymer expansion in carbon black–polymer composites [11–13]. However, besides the particular microscopic mechanism of resistance change, the very existence of an asymptotic value of the final resistance and the possibility of resistance recovery (reversibility) imply that two competing processes, lowering and enhancement of the local resistance, lead together to the trimming phenomenon.

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Network model of voltage trimming

In this paper, we report on our study of the trimming phenomenon, where we have mainly focused our attention on a statistical description of the process rather than on the actual microscopic mechanism of local resistor changes. We model the system as a regular lattice made of random resistances in which the trimming field is a constant high voltage pulse. The evolution of the local elements is governed by quite general local properties modelled using joule heating and dielectric breakdown. We show that various properties observed in experiments can be understood by requiring two competing behaviours of the local elements: a lowering and an enhancement of the local resistance.

2. Random resistor network We start our analysis by first considering a random resistor network model for the resistor prior to trimming. Let us consider a D-dimensional cubic lattice whose bonds have random conductance values g. The simplest interesting case is that of a bimodal distribution ρ0 (g) of bond conductances in which a fraction 0 ! p0 ! 1 of bonds have g = g1 and the remaining fraction 1 − p0 has g = g2 : ρ0 (g) = p0 δ(g − g1 ) + (1 − p0 )δ(g − g2 ).

(1)

When g1 > g2 , the macroscopic conductance, G, obtained by applying a voltage difference between two opposite ends of the lattice decreases monotonically when p0 goes from p0 = 1 down to p0 = 0. The special case g2 = 0 corresponds to the classical random resistor percolation problem, in which for p0 smaller than a critical value pc (pc = 0.5 for D = 2 and pc " 0.2488 for D = 3) the macroscopic conductance is zero. We can imagine that under the influence of an applied high voltage difference, the bond conductances will change their values according to some probability. More specifically, we consider the case in which a bond with g = g1 is changed into a g = g2 bond with probability W12 . Conversely, a g2 bond is turned into a g1 bond with probability W21 . In general, W12 and W21 depend on the trimming voltage and the microscopic properties of the compound. Moreover, in principle, such probabilities will also depend on the duration of the applied voltage and its profile. However, in the following we will not be concerned with such details and will treat the trimming process as being due to a series of voltage pulses of fixed duration, separated from each other by time intervals sufficiently long to permit the system to relax before a subsequent voltage pulse is applied. In this paper, we shall consider two different situations. In one case we consider W12 and W21 as truly random variables independent of the particular state of the network. In this situation, the network evolution is basically identical to a standard percolation problem. A second, more realistic, case is that in which the probabilities W12 and W21 depend explicitly on microscopic properties such as the local joule heating or the local electric field. The resulting network evolution is that of a biased percolation, where the induced change in bond conductances depends on the state of the network.

2.1. Standard percolation Let us consider the probabilities W12 and W21 as independent of the state of the network. This situation has already been studied in [14] with respect to the conductance evolution towards electrical breakdown. Here we propose a slightly different analysis aimed at describing trimming-like phenomenon. Let p0 be the initial fraction of bonds with conductances g = g1 , so that the initial macroscopic conductance is Gin = G(p0 ). By applying a first voltage pulse, we assume that the fraction of bonds that initially had g = g2 and has been converted to g = g1 is (1 − p0 )W21 , while p0 (1 − W12 ) is the fraction of the g = g1 bonds that have not changed their state. Hence, under the effect of W12 and W21 the resulting fraction, p1 , of g1 bonds is p1 = p0 (1 − W12 ) + (1 − p0 )W21 . The sequence is then repeated, and at the nth step (or equivalently at the nth voltage pulse) one obtains pn = pn−1 (1−W12 )+(1−pn−1 )W21 , where pn−1 is the fraction of g1 bonds resulting after n−1 subsequent voltage pulses. Hence, it is easy to obtain pn = p0 (1 − W12 − W21 )n +

W21 [1 − (1 − W12 − W21 )n ] . W12 + W21 (2)

The asymptotic regime is obtained by the n → ∞ limit, which from the above expression is p = lim pn = n→∞

1 1 + W12 /W21

.

(3)

Depending on the value of W12 /W21 , the asymptotic fraction, p, of bonds with conductances g = g1 can be larger or smaller than the initial value, p0 . Therefore, when g1 > g2 , the final macroscopic conductance Gfin = G(p) is larger than Gin if p > p0 , or, conversely, Gfin < Gin if p < p0 . To discuss the resistance evolution in more quantitative terms, let us consider the effective medium approximation of a RRN on a square lattice at the nth step of the sequence [15,16], ! Gn − g = 0, (4) dg ρn (g) g + Gn where ρn (g) = pn δ(g − g1 ) + (1 − pn )δ(g − g2 ) is the bond conductance distribution function at the nth step of the trimming process and Gn is the effective lattice conductance. The solution of equation (4) is " (2pn − 1)(g1 − g2 ) (2pn − 1)2 (g1 − g2 )2 + + g 1 g2 , Gn = 2 4 (5) where pn is given by equation (2). In figure 1 we plot equation (5) for g1 = 1, g2 = 0 and different values of the initial bond probability, p0 . In this example, we have set W12 = 0.05 and W21 = 0.2, corresponding to p = 45 = 0.8. As expected, for p0 < p (p0 > p) the macroscopic conductance, Gn , increases (decreases) monotonously with n and reaches the asymptotic value G(p) = 35 = 0.6 regardless of the initial condition. In order to check the consistency of the analytical results we have also performed Monte Carlo calculations on a 100 × 100 square lattice, and the resulting values of Gn are plotted in figure 1 as open and solid symbols. In solving the Kirchoff equations, we have employed the conjugate gradient method [17], and each 2171

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Figure 1. Conductance evolution for standard percolation with W21 = 0.2, W12 = 0.05 and different values of the initial bond probability, p0 . The bond conductances are g1 = 1 and g2 = 0. Symbols: Monte Carlo calculations of a 100 × 100 square lattice. Lines: effective medium approximation results (equations (2) and (5)).

Figure 2. Conductance evolution for standard percolation with WR = 0.2, p0 = 0.8 and different values of WD . The bond conductances are g1 = 1 and g2 = 0. Symbols: Monte Carlo calculations of a 100 × 100 square lattice. Lines: effective medium approximation results (equations (2) and (5)).

symbol in figure 1 is the average value of Gn for 50 different runs. The agreement with the analytic results (solid lines) is satisfactory and indicates that equations (5) and (2) give a rather good description of the conductance evolution in the standard percolation case. In figure 2 we show analytic and Monte Carlo calculations for the case in which the starting bond probability is fixed at p0 = 0.8 and the probability W12 is varied. As in the previous example we have set W21 = 0.2, g1 = 1 and g2 = 0.1. This case shows how the asymptotic regime is governed by W12 /W21 , i.e. the final conductance can be higher or lower than the initial one, depending on the ratio W12 /W21 . As we have assumed, if W12 and W21 are governed by an applied voltage pulse, then the above examples show in a simple way how the trimming phenomenon can be interpreted as a competition between two opposite processes and how the asymptotic regime is a result of a balance between the two probabilities W12 and W21 . Such a simple picture can also 2172

Figure 3. (a): Conductance as a function of the initial bond probability, p0 . Curve labelled with 1: g1 = 3, g2 = 0.1. Curve labelled with 2: g1 = 1, g2 = 0.01. The vertical dashed line denotes the asymptotic bond probability p = 0.6, while the horizontal dashed line indicates G = 0.6 for both curves 1 and 2. (b) Conductance evolution for the situation depicted in (a) with WR = 0.2 and WD = 2WR /3 " 0.133. Symbols are Monte Carlo calculations for a 100 × 100 square lattice, while the solid lines are the corresponding effective medium results.

account for more complicated situations like that described in figure 3. In panel (a) we plot the conductances of two different systems as a function of the initial bond probability. System 1 is characterized by g1 = 1 and g2 = 0.1, while system 2 has g1 = 3 and g2 = 0.01. Let us suppose that initially the two systems have the same macroscopic conductance as that given by the intercept between the dashed horizontal line and curves 1 and 2. In the example of figure 3(a), an initial conductance G0 = 0.6 corresponds to p0 = 0.5 and p0 = 0.8 for systems 1 and 2, respectively. If we suppose that both systems have W12 = 0.2 and W21 = 0.1, then the asymptotic bond probability will be p = 0.7, which lies between the p0 values of the two systems. Since the asymptotic regime is the same, the conductance evolution of 1 and 2 follows the paths shown in figure 3(a) as thick curves. For this particular situation, the resulting analytical and numerical conductance evolutions are plotted in figure 3(b), where it is shown how system 1 and system 2 depart from the initial conductance value to reach their respective asymptotic regimes, characterized by different values of g1 and g2 , but with equal asymptotic bond probability, p. The standard percolation model just discussed is certainly too oversimplified to describe quantitatively the trimming process; nevertheless it has the important merit of displaying many of the phenomena characteristic of trimmed resistors at least as long as conductance evolution is concerned. This is not a coincidence, but it is merely due to the underlying assumption that the trimming process is governed by two opposing processes, here played by W12 and W21 . In fact, as we show in the next session, more realistic models in which the local resistance change depends on the state of the network display properties very similar to those of standard percolation. 2.2. Biased percolation The microscopic processes governing the trimming phenomenon are of different origin, depending on the particular system considered. For example, in polysilicon resistors it

Network model of voltage trimming

is believed that the local joule heating can be so important as to trigger structural changes (melting) in the highly resistive regions separating adjacent silicon crystals [3]. Electromigration, healing of bad junctions and dielectric breakdown are just a few of the microscopic processes called into play in describing the trimming process. Despite such variety, in all these processes the local variations of conductances depend on the local state of the system. For example, the joule heating depends on the local currents and conductances. Hence, a description of the trimming phenomenon more realistic than that in the previous discussion should take explicitly into account local properties of the network [18, 19]. Here, we consider two different models constructed in order to include local variables and to describe a variety of trimming processes. 2.2.1. Model A. In the first model (model A) we assume that the bond conductances are distributed according to equation (1) with g1 & g2 > 0. The local variable is the bond temperature, Ti , defined as Ti = T0 [1 + Agi (#Vi )2 ],

Figure 4. Monte Carlo calculations on a 100 × 100 square lattice for model A. (a) g1 = 1, g2 = 0.1, E21 = 10, E12 = 7, E = 8. (b) p0 = 0.8. Model B with parameters g1 = 1, g2 = 0, E21 = 10, E12 = 7, E = 7.

(6)

where #Vi is the potential drop along bond i with conductance gi and A is a constant. The above equation is just the bond temperature variation driven by the joule heating. Further, we assume that the local temperature governs the probabilities of the bond conductor changes according to the following rules: W21 = e−E21 /T ,

(7)

W12 = (1 − e−E12 /T )e−E12 /T ,

(8)

where E12 and E21 , with E21 > E12 , are suitable activation energies. Equations (6)–(8) have been modified from those appearing in [19] in order to favour the recovery process of bad conductors (g2 → g1 ) at high temperatures (T > E21 ) and to favour the damage of good conductors (g1 → g2 ) at sufficiently low temperatures (T < E12 ). This particular form of W21 and W12 has been chosen in order to mimic the behaviour of polysilicon resistors under the trimming process [3]. Of course, as long as T ' E12 , E21 the system is frozen and no changes are permitted. We have solved numerically the above model on a 100 × 100 square lattice by applying suitable values of an external electric field E. The results for g1 = 1, g2 = 0.1, E21 = 10 and E12 = 7 are shown in figure 4. In figure 4(a) we have set the electric field equal to E = 8 for several values of the initial bond probability, p0 . As already discussed for the standard percolation model (figure 1) the asymptotic regime is independent of the initial bond probability, p0 , so that Gin = G(p0 ) can be higher or lower than Gfin . In figure 4(b) results are reported keeping p0 = 0.8 and for different values of the applied electrical field, E. The qualitative similarity between this biased percolation and the standard percolation model of the previous section indicates that the basic physics underlying the trimming process governed by the local temperature changes is very simple: it is merely the network evolution towards less or higher disorder than the initial one.

Figure 5. Monte Carlo calculations on a 100 × 100 square lattice for model B. (a) Model B with parameters g1 = 1, g2 = 0, E21 = 10, E12 = 7, E = 7. (b) p0 = 0.8.

2.2.2. Model B. When g1 (= 0 and g2 = 0, as in disordered conductor–insulator composites, there is no joule heating effect on the insulating bonds (Ti remains equal to T0 ), and a different mechanism of resistor recovering should be called into play. Hence, we introduce a different model that is basically a variation of model A in which the recovery process W21 is not governed by temperature changes as in equation (7) but rather by a sudden change driven by the local potential difference. This is somehow in the same spirit of a dielectric breakdown phenomenon in which an insulating bond becomes conducting as soon as the potential drop along the bond becomes larger than a given threshold E21 . Results of this model are shown in figure 5 for g1 = 1 and g2 = 0 and for the same values of E12 and E21 of model A. In figure 5(a) we have set E = 7 and different values of the initial bond probability, p0 , while in figure 5(b) we have set p0 = 0.8. Also in this case, the overall behaviour is very similar to the one obtained for the simple standard percolation model, confirming once again that the competition between recovering and damaging of local resistors governs the global behaviour. 2173

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3. Conclusions In the above study we have shown that a complete description of the trimming phenomena in disordered conductors must consider the statistical properties of the system as a whole. We have studied the trimming phenomenon within a statistical description of the process rather than on the actual microscopic mechanisms of local resistor changes. We have shown that various properties observed in experiments can be understood by requiring two competing behaviours of the local elements, a lowering and an enhancement of the local resistances. This scenario is already qualitatively captured by a standard percolation model in which the probabilities of lowering and enhancement of the local resistors are independent of the state of the system. More realistic models in which the local changes depend upon microscopic properties such as the bond joule heating or the voltage drop along the bonds confirm the simple standard percolation picture. We expect that the qualitative agreement between standard and biased percolation holds true as long as the system is far away from a critical region, such as that arising from a macroscopic electrical failure.

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