Black's Equation is a mathematical model for the
mean time to failure (MTTF) of a semiconductor circuit due to
electromigration: a phenomenon of
molecular
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
rearrangement (movement) in the
solid phase
In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetizati ...
caused by an
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
.
The equation is:
[
R. L. de Orio]
Dissertation: "Electromigration Modeling and Simulation"
2010.
R.L. de Orio, H. Ceric, S. Selberherr
"Physically based models of electromigration: From Black’s equation to modern TCAD models"
Microelectronics Reliability journal.
2010.
is a constant
is the
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ar ...
is a model parameter
is the
activation energy
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules pe ...
is
Boltzmann's constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
is the
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
in
K
The model is abstract, not based on a specific physical model, but flexibly describes the failure rate dependence on the temperature, the electrical stress, and the specific technology and materials. More adequately described as descriptive than prescriptive, the values for ''A'', ''n'', and ''Q'' are found by fitting the model to experimental data.
The model's value is that it maps experimental data taken at elevated temperature and electrical stress levels in short periods of time to expected component failure rates under actual operating conditions. Experimental data is obtained by running a combination of
high temperature operating life
High-temperature operating life (HTOL) is a reliability test applied to integrated circuits (ICs) to determine their intrinsic reliability. This test stresses the IC at an elevated temperature, high voltage and dynamic operation for a predefined pe ...
(HTOL), electrical, and any other relevant operating environment variables.
References
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{{DEFAULTSORT:Black's Equation
Electronic design automation
Electronics concepts