Chapter 9 Reliability
9.2 Fundamentals of Reliability Methodology
Reliability design based on well-designed, well-understood, and thoroughly implemented accelerated tests, therefore, is a critical part of microsystems packaging. In general, reliabil-ity design addresses questions concerning when failure occurs, what causes failure, how to avoid failure, and what is the expected lifetime of an electronic device.
The failure rate is often used as a general index for representing semiconductor reliability.
Semiconductor failure rates follow a bathtub curve as shown in Figure 9.1. The curve can be divided into three regions of initial failures, random failures, and wear-out failures according to the time of occurrence.
Time
Failure rate
Initial failure
period Random failure period
Wear-out failure period
Figure 9.1 Semiconductor failure rates in a typical bathtub curve
1. Initial Failures
These are failures that occur within a relatively short time. Therefore, these failures are also called infant mortality failures, and this period is also called early failure period. They can be characterized by a decrease in the failure rate over time. The main causes of infant mortality failure are manufacturing or material defects. For example, a defect can be caused by tiny particles in a chip during the device production process, resulting in a device failure later. Because only devices with quality defects will fail early, removal of these failed devices will decrease the failure rate.
2. Random Failures
These are failures that occur at a fairly constant failure rate over a long period of time in between the early failure period and the wear-out failure period. After the devices with quality defects have been removed, the remaining high-quality devices operate stably. In this period, the failures can usually be attributed to randomly occurring excessive stress, such as power surges, other high-energy radioactive rays, etc.
3. Wear-out Failures
These are failures caused the aging of devices from wear and fatigue and occur due to the physical limitations of the materials that comprise semiconductor devices. The failure rate tends to increase rapidly in this period, and these failures are used to determine the device’s lifetime. Therefore, the products are designed so that wear-out failures will not occur within their guaranteed lifetime.
Accordingly, it is important to reduce the initial failure rate to ensure long lifetime and durability against wear-out failures. Traditionally, individual companies implement quality controls and improvements as well as screening strategies, including electrical characteristics testing and burn-in tests. Furthermore, design-for-reliability is widely considered during the design and development stages, instead of the manufacturing stage alone.
9.2 Fundamentals of Reliability Methodology 175
9.2.1 Basic Reliability Theory[1,3]
The objective of any reliability study is to produce safe and reliable products. The eval-uation and quantification of product reliability are prerequisites for reliability improvement and are necessary for determining trade-offs between reliability improvement and cost during design. Four measures are often used to quantitatively represent reliability. Their definitions are described below.
1. Cumulative failure distribution function (CDF): F (t)
CDF is the proportion of components, devices, parts, or elements that cease to perform their designed functions after being used for a period of time t. This can be expressed by the equation
F (t) = c(t)
N , (9.1)
where c(t) is the number of failures to develop up to time t, and N is the total number of tested components.
2. Failure probability density function (PDF): f (t)
f (t) denotes the probability of a device failing in the time interval dt at time t. From this definition PDF can be expressed by the equation
f (t) =dF (t)
dt =−dR(t)
dt . (9.2)
As can be seen from the above equation, R(t) and F (t) can be calculated by taking the integral of the PDF, as below:
F (t) =
t
0
f (t)dt, (9.3)
R(t) = 1−
t
0
f (t)dt =
∞
t f (t)dt. (9.3a)
3. Reliability function: R(t)
The reliability function is the proportion of components (devices, parts, or elements) that continue to perform their designed functions and remain stable after time t. From the PDF definition, it is easy to get the reliability function as
R(t) = 1− F (t). (9.4)
The reliability function R(t) is a monotonically decreasing function, and the F (t) is a monotonically increasing function.
4. Failure rate: λ(t)
The failure rate λ(t), also known as “hazard function,” represents the probability of failure occurring in the next time interval for devices that have not yet failed when time t has passed. Using the concept of conditional probability, P (B|A) = P (B and A both occur)
P (A) ,
we can derive that λ(t) equals f (t)/R(t) as shown below:
λ(t) = f (t)
R(t). (9.5)
176 Chapter 9 Reliability
The reliability function R(t) can be expressed by λ(t) as follows:
R(t) = exp
5. Mean time to failure (MTTF)
In general, once a component has failed, it cannot be repaired and used again. That is to say, it is a nonreparable component. MTTF is a basic measure of reliability for this type of device. It is the mean time expected until the first failure occurrence. MTTF is a statistical value and is meant to be the mean over a long period of time and a large number of units.
It can be given by
MTTF =
∞
0
tf (t)dt (9.7)
The overall relationship of F (t), R(t), f (t), λ(t), and MTTF can be depicted as Figure 9.2.
R(t)
Figure 9.2 Overall relationships of basic reliability measurements
9.2.2 Distribution Used in Reliability Analysis
Since reliability stress tests are often destructive, only a sample population is used for reliability testing. As such, the assessment of reliability for the rest of the population is es-sentially statistical and probabilistic in nature. We will discuss commonly used distributions for semiconductor device reliability in detail.
1. Weibull Distribution
The Weibull distribution is a highly general-purpose distribution function that is expanded from the logarithmic distribution. In reliability data analysis, this model is frequently used to analyze life data in reliability tests, etc. The probability density function f (t) and distri-bution function F (t) of this distridistri-bution are as follows.
The three-parameter Weibull PDF is given by f (t) = β parameter (or slope), γ= location parameter.
We know through experience that no failures occur before a given test time γ, and after γ the total number of failed devices increases with time t (or more correctly, maintains a non-decreasing trend). The location parameter γ = 0 if we assume that the probability of failure is already above 0 immediately before testing. The three-parameter Weibull distribution will reduce to the two-parameter Weibull distribution, and its PDF is given by
f (t) = β
9.2 Fundamentals of Reliability Methodology 177 Figure 9.3 shows the effect of different values of the shape parameter β on the shape of the PDF. One can see that the shape of the PDF can take on a variety of forms based on the value of β.
Figure 9.3 The effect of different values ofβ on the shape of the Weibull distribution’s PDF
The equation for the three-parameter Weibull cumulative density function F (t) is given by
F (t) = 1− e−(t−γη )β. (9.10) The reliability function R(t) for the three-parameter Weibull distribution is given by
R(t) = e−(t−γη )β. (9.11)
The Weibull failure rate function λ(t) is given by λ(t) = f (t)
R(t) = β η
t− λ η
β
. (9.12)
From Figure 9.4, we can see that the function form of the Weibull distribution is capable of representing different failure modes depending on the value of the parameter β. When β = 1, the PDF of the three-parameter Weibull reduces to that of the two-parameter exponential distribution as
f (t) = 1
ηe−t−γη , (9.13)
Figure 9.4 Effect of the shape parameterβ on the failure rate of the Weibull distribution
178 Chapter 9 Reliability where λ = 1
η is a constant.
When β > 1, the Weibull failure rate λ(t) monotonically increases, representing a wear-out failure mode. For 1 < β < 2, the λ(t) curve is concave, consequently, the failure rate increases at a decreasing rate as time increases. For β = 2 there emerges a straight line relationship between λ(t) and t, starting at a value of λ(t) = 0 at t = γ, and increasing thereafter with a slope of 2
η2. Consequently, the failure rate increases at a constant rate as time increases. Furthermore, if η = 1 the slope becomes equal to 2, and when γ = 0, λ(t) becomes a straight line that passes through the origin with a slope of 2. Note that at β = 2, the Weibull distribution equations reduce to that of the Rayleigh distribution. When β > 2, the λ(t) curve is convex, with its slope increasing as time increases. Consequently, the failure rate increases at an increasing rate as time increases, indicating wear-out life.
When β < 1, the Weibull failure rate λ(t) is unbounded at t = 0 (or γ) and decreases thereafter monotonically and is convex, approaching the value of zero as t→ ∞ or λ(∞) = 0.
This behavior makes it suitable for representing the failure rate of units exhibiting initial failures, for which the failure rate decreases with increasing time. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping.
2. Normal Distribution
The normal distribution is commonly used for general reliability analysis and times-to-failure of simple electronic and mechanical components, equipment, or systems.
The PDF of the normal distribution is given by f (t) = 1
σ√
2πe−12(t−μσ )2, f (t) 0, −∞ < t < ∞, σ > 0,
(9.14)
where μ is the mean of the normal times-to-failure and σ is the standard deviation of the times-to-failure.
The normal distribution is symmetrical about its mean value, and the standard deviation σ is the scale parameter of the PDF. As σ decreases, the PDF becomes narrower and taller.
As σ increases, the PDF becomes broader and shallower, as shown in Figure 9.5. The
Figure 9.5 The typical normal distribution PDF with different standard deviation
9.3 Wafer and Packaging-related Failure Mode and Mechanisms 179