Arrhenius Equation in Chips: Acceleration Factor Explained

In everyday life

Food keeps longer in the fridge. The chemical reactions and microbes that spoil it work more slowly at lower temperatures. On a hot summer day, the same food spoils much sooner.

Semiconductor chips also wear out slowly over time. But in normal use, it takes years for them to fail. So makers run chips nonstop at around 125 °C, deliberately speeding up the wear, and study the result. The Arrhenius equation is how they work out how many years at normal temperature 1,000 hours at high temperature are worth.

In chip factories, too, temperature is an important "dial." Growing films on the silicon surface and spreading added atoms are done in furnaces at around 1,000 °C. A small change in temperature makes a big change in the finished thickness or depth.

How it works 1: the rate depends on how many particles get over the barrier

The atoms and molecules in matter are always jiggling or flying around. Temperature is a measure of how strong this jiggling is. But the particles do not all move with the same energy. Some are slow, and some are very energetic.

For a reaction to happen, or for an atom to hop to the next site, it must pass through a high-energy state on the way. It is like having to climb a hill to reach the next valley. The height of this hill is called the activation energy (Ea). Only a tiny share of particles, the ones that happen to have a lot of energy, get over it.

The hill's height is often given in a unit called the eV (electron volt). 1 eV is the energy one electron gains when pushed by 1 volt. For wear-out and reactions in semiconductors, hills range from a few tenths of an eV to a few eV.

Temperature here is not in everyday °C but in absolute temperature. Absolute temperature is the °C value plus 273.15, in units of K (kelvin). The scale puts 0 K where particles jiggle the least, so it is roughly proportional to how strong the jiggling is.

A rough measure of the size of thermal jiggling is "kT," absolute temperature T times a constant k. Here k is the Boltzmann constant (about 8.617×10⁻⁵ eV/K), a factor that converts temperature into energy. At 25 °C (298 K), kT is about 0.026 eV.

What matters is the ratio: how many times kT is the hill's height? According to the physics of heat (statistical mechanics), each extra kT of energy cuts the number of particles that reach it to about 1/2.7. If the hill is 3 times kT, multiply by 1/2.7 three times, and you get about 1/20. To keep things simple, we leave out smaller factors here.

Because the same fraction is multiplied again and again, a few more steps change the result by factors of ten. This is how exponential functions behave. The number 2.7… is called e, and "e to the power −3" is written exp(−3). The rough share that gets over, exp(−Ea/kT), is called the Boltzmann factor.

For example, take a hill of 0.7 eV. At 55 °C, the hill is about 25 times kT. That means multiplying by 1/2.7 about 25 times, and only about 1 particle in 56 billion gets over. At 125 °C, kT is larger, and the ratio drops to about 20. With just over 4 fewer multiplications, about 1 particle in 700 million gets over: about 78 times more. In absolute temperature, that is a rise of only about 20%.

How it works 2: reading the Arrhenius equation in words

A reaction's rate can be thought of as "how often particles try the hill" times "the share that gets over." How often they try does not change much with temperature. Almost all of the rate's change with temperature comes from the share that gets over: the Boltzmann factor.

Written as an equation, this is the Arrhenius equation. Let A be a factor that bundles together how often particles try, and other things. Then:

rate = A × exp(−Ea / kT)

Textbooks sometimes also use the letter k for the rate (the rate constant) on the left, writing k = A·exp(−Ea/kT). These two k's are different things; the one on the right is the Boltzmann constant.

The equation is named after the Swedish scientist Arrhenius. In 1889, in a paper on how acid breaks down sugar (sucrose), he explained what the equation means. Chemistry textbooks use energy per mole (a fixed, very large number of particles) instead of per particle, and write exp(−Ea/RT). It is the same equation.

Here, we treat A and Ea as not changing with temperature. This is the rule of Arrhenius's original form. In reality, A can change a little with temperature too, but that is small next to the change in the exponential part.

How it works 3: reading activation energy from an Arrhenius plot

You can find a reaction's activation energy by measuring its rate at several temperatures. Plot the inverse of absolute temperature (1/T) across, and the logarithm of the rate up. This graph is called an Arrhenius plot.

Plotted this way, the measured points fall on a straight line. The inside of the exponent is "−Ea/k × 1/T." So with 1/T across, and the logarithm undoing the exponential, what remains is just the equation of a straight line. The steeper the line, the larger the activation energy, and the more sensitive the reaction is to temperature.

Even without a graph, if you know the ratio of rates at two temperatures, you can work back through the calculation in the next section. For example, if the rates at 55 °C and 125 °C differ by about 78 times, the activation energy is about 0.7 eV.

Points that fall on one straight line suggest that a single mechanism sets the rate over the measured temperature range. If the points bend partway, a different mechanism has taken over. We use this later, in "When the equation does not apply."

By the numbers: how many years is 1,000 hours at 125 °C?

How many times faster wear-out runs at the test temperature than at the use temperature is called the acceleration factor. With the Arrhenius equation, it is the rate at the test temperature divided by the rate at the use temperature. A cancels out in the division, so only the activation energy and the two temperatures matter.

acceleration factor = exp[(Ea / k) × (1/T_use − 1/T_test)]

Let's assume an activation energy of 0.7 eV, and work out a chip used at 55 °C and tested at 125 °C.

1. Convert to absolute temperature: 55 °C = 328.15 K, 125 °C = 398.15 K
2. Difference of the inverses: 1/328.15 − 1/398.15 ≈ 0.000536 (1/K)
3. Ea/k: 0.7 ÷ 0.00008617 ≈ 8,120 (K)
4. Multiply them to get about 4.35. e to the power 4.35 gives an acceleration factor of about 78

So 1,000 hours at 125 °C equals the wear of about 78,000 hours, or about 8.9 years, at 55 °C. The temperature here is not the room temperature but the temperature inside the working chip (the junction temperature). A chip heats itself above its surroundings, so estimate it first with the junction temperature calculator.

As the table shows, the answer shifts by tens of times depending on the assumed activation energy. Which value to use depends on the cause of the wear (the failure mechanism). JEP122, a document from the industry group JEDEC, lists activation energies and models for each cause. When the cause is unknown, 0.7 eV is often used as a stand-in value, and makers' reliability reports include examples. It is only a stand-in.

Where it shows up in semiconductors

Thermal oxidation: how fast oxide grows

Put silicon in hot oxygen, and an oxide film (SiO₂) grows on its surface. This is called thermal oxidation. How the film grows is described by the Deal–Grove model of 1965. The model's two constants both follow the Arrhenius equation.

While the film is thin, the reaction at the boundary with the silicon sets the rate, with an activation energy of about 2.0 eV. As the film thickens, how fast oxygen seeps through the film takes over. For dry oxidation, grown in dry oxygen, that is about 1.2 eV. Going from 1,000 °C to 1,100 °C makes the first about 3.8 times faster and the second about 2.3 times. You can change the temperature and check with the thermal oxide thickness calculator.

Impurity diffusion: how fast atoms spread

Boron or phosphorus atoms added to silicon slowly move through the crystal at high temperature and spread out. Adding impurities by using this is called thermal diffusion. The diffusion coefficient D, which shows how easily atoms spread, also has the Arrhenius form: D = D0·exp(−Ea/kT). Typical values are fairly large: about 3.5–3.7 eV for boron and phosphorus, and around 4 eV for arsenic.

At 3.5 eV, going from 1,000 °C to 1,100 °C makes the diffusion coefficient about 10 times larger. So factories track the running total of time spent at each temperature (the thermal budget). They also use methods that heat for only a very short time (furnace anneal vs. RTA vs. laser anneal). Real diffusion also depends on impurity concentration and crystal defects, so this equation is a starting point.

CVD: reaction-limited vs. transport-limited

Depositing a film by reacting gaseous raw materials on a surface is called CVD (chemical vapor deposition). At fairly low temperatures, the surface reaction is slow, and its speed sets the deposition rate. This is called the reaction-limited regime. The rate follows the Arrhenius equation and is sensitive to temperature.

Raise the temperature until the reaction is fast enough, and now the gas cannot reach the surface fast enough. This is called the mass-transport-limited regime. Raising the temperature further barely changes the rate. On an Arrhenius plot, the line looks bent partway. Most low-pressure CVD runs on the reaction-limited side, where it is easier to keep film thickness even across the whole wafer.

Reliability tests: HTOL and electromigration

A test that runs chips nonstop at high temperature, with voltage applied, is called HTOL (high-temperature operating life). It follows JEDEC's standard JESD22-A108, and makers' reliability reports often show the condition 125 °C for 1,000 hours. The acceleration factor above is what converts the result into years at the use temperature.

Metal atoms in the wiring can be pushed along slowly by the flowing electrons, until a wire eventually breaks. This is called electromigration. Its lifetime is estimated with Black's equation. It multiplies a current-density term by a temperature term of the Arrhenius form. Its activation energy depends on the metal and on the paths the atoms take.

Leakage current: electrons made by heat

A small current leaks even through a transistor or diode that should be off. Part of it is carried by electrons and holes that heat creates in the silicon. This part also follows the Boltzmann factor, rising steeply with temperature. "What Is a Semiconductor?" explains how heat creates more electrons.

The "barrier" here is linked to the band gap (about 1.12 eV for silicon). Depending on how the leakage arises, the apparent activation energy is about half of that, or about the same (Sze's textbook). The number of electrons that heat creates in pure silicon (the intrinsic carrier density) rises about 67 times from 25 °C to 85 °C. You can check this with the intrinsic carrier density calculator. The part of the leakage proportional to this number grows by about as much. The part proportional to its square grows thousands of times.

When the equation does not apply

When the limiting step switches: as in the CVD example, if the mechanism that sets the rate changes across a temperature range, one straight line cannot describe it. Extend a line measured at low temperature to high temperature, and it misses.

When there is more than one cause: a real chip has several causes of wear-out, each with its own activation energy. The total failure rate is estimated by adding up the failure rate of each cause. Lumping them into one value can be far off at some temperatures.

When factors other than temperature matter: humidity and voltage also speed up wear-out. Other models are combined with it, such as Peck's model for humidity, and Eyring-type models for voltage and other non-temperature conditions. An example of a test that adds humidity is HAST (highly accelerated stress test).

Common misconceptions

Key numbers

Learn more

• Terms in detail: Arrhenius model (the equation and the 10 °C rule) / bathtub curve (how failure rates change over time)
• Compare in a table: HTOL vs. THB vs. HAST vs. temperature cycling vs. HTS / TDDB vs. HCI vs. NBTI / LPCVD vs. PECVD
• Try the numbers: thermal oxide thickness / junction temperature
• The whole process: "How Are Chips Made?"