ABSTRACT
Reliability theories and testing are based on the fact that most chemical changes that occur in devices are governed by the Arrhenius equation, whereby the rate of change varies exponentially with temperature (T): R = A E
kT exp
a-ÏÌ Ó
¸ ˝ ˛
(22.1)where R is the reaction rate, A is the proportionality constant, sometimes a slow function of temperature, Ea is the activation energy, and k is the Boltzman constant (8.6 × 10-5 eV/K). Different failure mechanisms have different activation energies; the mechanisms and the value of the activation energy associated with them will be discussed later. The exponential dependence on temperature makes it possible to test semiconductor devices in a reasonable time at elevated temperatures. The data from these tests can be used to predict the lifetime at the operating temperature. If a physical change leads to a failure that can be defined, for example, drop in b by a certain percentage for a heterojunction bipolar transistor (HBT), in time t (assuming a constant reaction rate), at temperature T t
R = 1 (22.2)
Rearranging the Arrhenius Eq. 22.1 t = 1 A
Ê ËÁ
ˆ ¯˜ exp
a E kT
Ï Ì Ó
¸ ˝ ˛
(22.3) or ln (t) = C E
kT + a (22.4)which leads to ln t
t E k T T
1 1Ê ËÁ
ˆ ¯˜ = Ê ËÁ
ˆ ¯˜
- Ê ËÁ
ˆ ¯˜
(22.5) To accurately predict lifetimes at normal temperature, at least three high-temperature tests are conducted. The median life from each of the tests is transferred to an Arrhenius plot, which has log time as the x axis and 1/T as the y axis. A straight-line fit to this data is then used to project the lifetime at any temperature. Most reactions are thermally activated, and activation energy measures how rapidly the mean lifetime changes with temperature. Plotting failure data on a log time versus 1/T graph, Ea can be computed. A high activation energy does not necessarily mean a better or more reliable device. Figure 22.2 shows the lifetime as a function of temperature for different activation energies [1]. In this example, lifetimes below 150°C are longer for a high activation energy, but above 150°C, the reverse is true. At very high temperatures, the mechanism of degradation itself may switch, leading to different activation energies. This can be seen as a change of slope of the lifetime versus 1/T data. Accelerated life tests should be done with large sample sizes so that the lifetime can be predicted with a high degree of confidence. In practice, a more practical or economical sample size is chosen. For a large group being subjected to accelerated testing, devices fail at different times. The median life is the time at which 50% of the devices have failed. Failures are going to follow a statistical distribution function, and testing must be continued until 50% failures have accumulated. Semiconductor devices almost always follow a lognormal distribution, among a variety of other distributions, for example, normal, Weibull, exponential, etc. For normal distribution, a plot of the logarithm of time to failure as a function of percentage cumulative failure is a straight line. The cumulative distribution function, F(t) gives the distribution of cumulative failures over time.
