VOLTAGE DEGRADATION MODEL OF THIN FILM CAPACITORS

A degradation model of thin film capacitors is presented. This model takes into consideration that: 
(a) the damage rate dD/dt is a function of the damage value D, and (b) the critical damage Dc is a 
function of working voltage. On the base of this model, the short term breakdown voltage and its 
distribution is defined. The experimental data presented conforms with the described model.


INTRODUCTION
Usually in considering the reliability of capacitors the life time in the form of a power law 1,2'3 B (I) V n or an exponential law 3,4,s,6 r B: exp(-aV) (2) is assumed.In the above relations, v is the life time, V is the working voltage, B1, B2, a are constants.The relations (1) and (2) result from following degradation modeP dD s(v) (3) dt where D is the damage, S(V) is the damage rate as a function of working voltage.
The parameter D is a critical parameter of the capacitor, which due to physical or chemical aging processes changes as a function of time. 7,s When the damage D changes from its initial value Do to the critical value De, breakdown occurs.The value of the initial damage Do is a random variable and hence the life time and the breakdown voltage are random variables too.Typically for the breakdown voltage the Weibull type distribution is accepted ,9, p(Vb) 1 exp - (4)   where V b is the breakdown voltage, bl, X are the Weibull distribution parameters.
The relation ( 4) is defined for Vb from 0 to but from the experimental data the breakdown voltage should be limited to 0 to Vb max.Vb max is a maximum value of the breakdown voltage equal to the breakdown voltage of a defect-free capacitor.If Do--De, then the life time is equal to zero.This case takes place, when the working voltage is equal to the breakdown voltage.The above observations do not conform with relations (1), ( 2) and (4).Hence a correction of that underlying model is needed.This is particularly important in the case, when the elements are tested by using the accelerated methods (step or ramp test).In these methods the capacitors are tested at voltages near the breakdown voltage, when the relations (1) and (2) are not correct For instance applying a linear voltage increase Vbr B r n+ (5) where Wbr is the breakdown voltage for linear voltage increase, r is the rate of voltage increase, B and n are constants.From (5) one can find, that Wbr if r and this is not correct in a physical sense.

THE DEGRADATION MODEL
If one assumes, as in 7,8 the degradation model in the form: and that the function K(D) is in the form K(D) aD m (7) then the life time is given by:r S(V) D. K(D) a (m-l) S(V') Do m-Dc m- where a and m are constants.
The value of the critical damage is dependent on the working voltage.For the breakdown voltage, the value of initial damage Do is already a critical value Do--Dc.The value of Dc increases as the working voltage is decreased.If we assum that where dc and k are constants, then dc(m_l where 1 b (m-l)' kl k(m-1) al (m-l) dc (10) If the function of S(V) is given by ( 2) or (3) a relation for life time versus voltage may be obtained from (9).
In the above examples the initial damage and life time are the random variables.From (10) the life time can be defined, when fraction W of capacitors is broken [for W 50% the median life time is (v(50%)].From the value of Do, the short term breakdown voltage can be derived from the value of Do.This is the voltage at which the value of Do is critical, and now v O.
and for power or exponential relation of S(V) we have (1) or (2) respectively.
Eq. ( 12) is a generalization of the relation (1) or (2).The breakdown voltage Vb is a theoretical value.The breakdown voltage is measured at the increasing voltage and during the measurement the coefficient D changes.
If a linear voltage increase is assumed V=rt where r is a constant then where Vr is the life time with the linear voltage increase.From (7) The probability P(Vb), that the capacitor breaks at the votlage V < Vb is defined by the capacitor area and the density of weak spots.
where A is a capacitor area, '(VD) is the density of weak spots as a function of breakdown voltage.If we assume that the following relation approximately holds ,(Vb) M (20) Do where M and are constants and g b is a function of Do [eg. (11)]then M Vb max 1 (21) then (21) approaches the Weibull type relation. 9 '1   From (12) and (21) the life time distribution results.In Vb max > Vb and Vb k > Vk a Weibull type distribution is obtained:- particular if (22) This degradation model is compatible with the physical model described in. 1 It has been stated that a metallic spike is formed in the weak spot as a result of ionic current flow.The electric field at this spot is considerably higher than the average field as in the case of the edge effect.The rate of the spike height growth vs electric field is given by the approximate relational: dh b2 j(E)h 0 (23) d--where h is the height of the spike, j(E) is the ionic current and b2 is a constant.If it is assumed, that D h, then Eq. ( 23) is conformable to the Eqs.( 6) and (7).The local field El at the spike is E, E(1 + ah0) (24) The growth of the metallic spike lasts until the local electric field exceed the critical value, Ec, and then E E(1 + mh{) (25) and for the initial value of ho where E V/d, d is thickness of the dielectric, E b is a breakdown field (E b Vb/d), ho, hc are the initial and critical value of the spike height respectively.
From ( 25) is he ---1 (27 The above relation is identical to Eq. ( 9) if Ee Vb max E V

RESULTS OF THE MEASUREMENTS
Thin film capacitors A1-A1203metal were used in the epxeriments.The metallic layers were prepared by evaporation onto a Coming 7059 glass substrate.The A1203 layer was produced by anodic oxidation of aluminium in a solution of ammonium pentaborate in ethylene glycol.The thickness of the A12Oa film was little influence on the breakdown voltage.The measurement data of the breakdown voltage distribution is shown in Figure 2, with a rate of voltage increase of volt/sec.
Conformity with relation (21) was verified by plotting the results with ordinates scaled: In (-In (l-p)) In (Vb max/gb 1).It was assumed that Vb max--120 volts.
A straight line in the entire range of Vb is obtained.If this data is plotted in typical Weibull scale (ln(-ln(1-p)) In Vb) an aberration from straight line for voltage near Vb max is noticed.

CONCLUSION
The present model of the degradation of capacitors is a correction to the known model.This correction gives a significant change in a dependence of the life time on the working voltage, when the voltage is near the breakdown voltage.If the voltage is low, and V b >> V the relation remains unchanged in comparison with (1)   or (2).A similar situation occurs in the case of the breakdown voltage distribution.
When the breakdown voltage is Vb ' Vb max the Weibull type of distribution is obtained, but at voltages near Vb max the present model gives values closer to those observed.If the experimental data is presented on the scale ln(-ln(1-p)) In V b, then at a breakdown voltage near the Vb max a deviation from a straight line is obtained.This means, that in this breakdown voltage range the experimental data do not conform with the Weibull distribution.In the case of the ln(-ln(1-p)) and In (Vv max/gb 1) scale, as in Figure 2, a straight line in the entire range of Vb is obtained.When the capacitors are tested by applying a ramp test, Vbr increases asymptotically to Vb as the rate of voltage increases.This is in contrast to the known model (Eq.( 5)), in which Vbr increases up to For the experimental data presented in this paper, Vbr Vb, when th r > volts/sec.

FIGURE 2
FIGUREThe dependence of the breakdown voltage vs the rate of voltage increase.
voltage breakdown voltage for linear voltage increase rate of voltage increase maximum value of the breakdown voltage fraction of broken capacitors B1, Bz, n, al, m, dc, k, klconstants spike electric field (E V/d) local and critical electric field initial and critical value of the spike height thickness of dielectric constants Vbr is the breakdown voltage for linear voltage increase.For a high rate of voltage o