Using Models to Predict Semiconductor Failures

A companion piece to "What Causes Semiconductor Devices to Fail?" from our November 1999 issue.

Staff -- Test & Measurement World, 11/1/1999 2:00:00 AM

"Models Predict Failure Rates," which was part of the article, "What Causes Semiconductor Devices to Fail?" by V. Lakshminarayanan (Test & Measurement World, November 1999) described how to use the Arrhenius equation to predict failure rates, and it also referred to other types of models. The author provided the following additional information about these models, which give you a feeling for the types of effects that you can model mathematically. These descriptions do not provide exhaustive explanations of these models, however. For that we refer you to the publications in the For Further Reading section.

Accelerated Testing Models

Accelerated testing helps engineers obtain information about the lifetime distribution of a component within a compressed time, thereby reducing the cost of testing. This type of testing applies higher-levels stimuli to the component, but the stimuli remain within the component's overall capability to withstand stress.

Another testing approach--accelerated testing--increases the usage of a component in a short time, thus increasing the possibility that it will wear out. For example, if a relay must operate 10 times a day in a certain application, and the required operating life is estimated at 10 years, the relay must operate 36,500 times. You can test a relay and estimate the life expectancy of a batch of the same type of relays by performing 1000 operations per day within 36.5 days.

In the case of semiconductor devices, you can accelerate life testing by applying temperature cycling or another stress such as humidity. Such testing of components can expose latent defects. And data obtained from such tests is useful for predicting the reliability of components during operation. In most cases, you specify acceptance limits for parameters such as voltage, current, temperature, and so on, used during testing. Accelerated tests apply temperatures in the range of 75° C to 225° C, depending on the failure mechanisms you want to test for, and depending on the type of device you are testing. In many cases, you will apply a nominal voltage to the component.

You may also want to test components under conditions of high humidity, in the region of 50% relative humidity (RH) to 90% RH, and under temperatures ranging from 85° C to 150° C. A standard combination used is 85° C and 85% RH. Combined humidity and temperature tests are useful for screening components, such as those used in electronic systems that will operate in coastal areas. These tests can stimulated failure mechanisms that arise from corrosion, metallic growth due to migration of ions, and so on.

Tests in which environmental stresses are applied in addition to temperature are known as Highly Accelerated Stress Tests (HAST).

The models used to model accelerated life-testing on electronic components include the Arrhenius eualtion, the Eyring equation, the Reich-Hakim equation, the Peck equation, and the Lawson equation. (Engineers often interchange the words model and equation when they refer to the relation of failure rates to empirical data.)

Arrhenius Model

The Arrhenius model describes the relationship between failure rate and temperature for electronic components. It derives from the observed dependence of chemical-reaction rates on temperature changes. You can use the model to estimate the reliability of electronic components using accelerated life testing. According to this model, the reaction rate is given by:

R = Ae^(-Ea/kT)

In which:

R = failure rate (or reaction rate)
A = empirical rate constant
E_a = activation energy (eV)
k = Boltzmann's constant (8.6 x 10-5 eV/K)
T = temperature in Kelvin (° C + 273.16° C)

You can apply this model under the following conditions :

1. The major stresses depend on temperature, or

2. The life-time of a component follows a log-normal distribution at all temperatures.

Different types of failure mechanisms have different activation energies (Table), but temperature is the factor that can accelerates the failure mechanism. For semiconductor devices, the temperature, T, represents the devices junction temperature during operation.

Table: Activation energies for failure mechanisms

Failure Mechanism	Activation Energy, Ea (eV)
Oxide defects	0.3 to 0.5
Bulk silicon defects	0.3 to 0.5
Corrosion	0.45
Assembly defects	0.5 to 0.7
Electromigration	0.6 (Aluminum line) 0.9 (Contact)
Mask or photoresist defect	0.7
Contamination	1.0
Charge injection	1.3

 

Eyring Model

The Eyring model improves on the Arrhenius model by taking into effect environmental stresses such as thermo-mechanical phenomena. In practice,stresses other than thermal can cause failures. According to the Eyring model, the median life of a device is given by the equation:

t _50% = A e^{[(E/kT)· F(V) · F(RH)]}

In this equation, F_(V) represents the stress factor for applied voltage, F_(RH) represents the stress factor for relative humidity, and A represents the total-acceleration coefficient. The other variables are the same ones used in the Arrhenius model. You can use the Eyring equation to model various failure mechanisms after you select the appropriate value of the activation energy from the values provided in the Table above. According to this model, the total-acceleration factor for a chemical process with an activation energy of 0.3 eV is given by:

A = e ^{[(-E/k)(1/T0-1/T) - B(1/RH0-1/RH)]}

In this example, B = 296 and RH₀ and T₀ equal the reference conditions of relative humidity and temperature respectively. MIL-HDBK-217F uses this model to estimate the temperature and stress factors for electronic components.

Peck Model

This model accounts for temperature and humidity stresses separately, and it assumes normalized humidity and temperature stresses of 85 ° C and 85 % RH, respectively. You then use these factors to obtain the overall acceleration factor. The following equation describes the model:

A = e^{[(-E/k){(1/TO) - (1/T)}(RHO/RH)n]}

TO and RHO are the reference temperature and relative humidity respectively, and n = 4.5 in this model.

Reich-Hakim Model

This equation models the median life of a component as a function of temperature and humidity:

t _50% = e ^{[-{A + B(T + RH)}]}

In this equation, T represents the temperature in ° C , RH is the relative humidity as a percentage, A is the acceleration coefficient, and B is a constant. The acceleration coefficient is calculated from:

A = e^{[(-E/k) {1/(T0 + RH0) - 1/(T + RH)}]}

Here E is the activation energy of 1.19 eV, and T₀ and RH₀ represent the reference temperature and humidity conditions.

Lawson Model

In this model the acceleration factor is assumed to vary linearly with temperature and non-linearly with humidity. A represents the acceleration coefficient, B represents the acceleration coefficients due to temperature, and C represents the acceleration coefficient due to humidity:

A = B · C

B = e^{[(-E/k){(1/(273 + T) - (1/(273 + 85)}]}

and

ln C² (RH) = D(85² - RH² )

In these equations, T is the temperature in °C, RH is the relative humidity , D is a constant, k is Boltzmann’s constant, and E is the activation energy. Because 85°C and 85 % RH iare considered standard combination test condition, use these values for temperature and humidity in the equations. For E = 0.4 eV and D = 5.57 x 10 ^-4 the model is fairly accurate for the case of metallic corrosion.

 

FOR FURTHER READING

Pollino, Emiliano, "Microelectronic Reliability: Integrity Assessment and Assurance," Artech House, Norwood, MA, 1989, pp. 361--399.

Ramakumar, R., "Engineering Reliability: Fundamentals and Applications," Prentice Hall, Englewood Cliffs, NJ, 1992, pp. 402--415.