Thermal expansion

Updated October 16, 2009

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New content for November 2009! Thermal expansion is expressed in parts per million per degree Kelvin (ppm/K), when the metric system is employed. Thermal expansion can be expressed as linear, or volumetric.

Before we get too far with this topic, we need to correct a common misconception about thermal expansion. Engineers often refer to TCE (temperature coefficient of expansion) when they mean thermal expansion. What's the difference, you ask?

The expansion characteristics (linear and volumetric) of materials always vary over temperature. In some case, the variation is little, in some cases it is substantial. The degree of variation depends on the material, as well as the temperature range.

Here's the linear expansion characteristic "alpha" of silicon, in parts per million per degree K (PPM/K), versus temperature. We found these data on Ioffe Institute's web site. The "CTE" could be any number between -0.5 and 4.5 depending on what temperature it is measured at. CTE is the value of alpha at a single temperature point.

So, can we please all stop talking about "matching CTEs" of materials, and say that we are trying to match their linear thermal expansion curves? Talking about CTE when you mean expansion is like referring to a map and and measuring the distance between points in miles per hour. CTE is an instantaneous rate, not a measurable (nor particularly useful) parameter. Ignoring expansion characteristics over temperature will sometimes get you into very bad trouble.

When you consider the stresses that are created when different materials are joined, you need to consider the temperature that they were bonded at. If the materials are joined using gold-tin eutectic solder, the expansion mismatch is "based" at 280 degrees C. Thus at 25C (or room temperature) you have to chase down the two curves to look at the total mismatch. Semiconductor materials such as InP, GaAs or silicon like to be put into compression rather than expansion stress, so it is a good idea to join them to a material that has a slightly more aggressive expansion characteristic. Or to put this in terms of CTE (which we just objected to!) you want the housing material to have a higher CTE than the semiconductor, not vice versa.

Here's another point. When we talk about expansion, there's linear expansion, and volumetric expansion. Usually we are talking about linear expansion. For a isotropic material that has equal linear expansion in all three axes (and not all materials do!), the volumetric expansion is 3X the linear expansion. You can prove this with a little calculus, but we won't bother here.

Here's a web site with a ton of TCE data. For expansion curves, check out the Ioffe Institute's web site.

Non-isotropic materials

Many soft substrates have non-isotropic thermal expansion properties. This is usually because base material has a very high linear expansion coefficient (such as PTFE which has 110 PPM/C at room temperature), and suppliers want to reduce it in the X and Y dimensions so you can mount the material to a metal that has lower expansion, and not have to worry about it delaminating when it is temperature cycled.

There are two primary ways to do this. One is to put place low-expansion fibers oriented randomly in the X and Y axis. A second way is to embed a low-expansion fabric in the material (in the X-Y axis). In ether case, you have to consider the dielectric properties of the fiber or fabric in the composite material.

The penalty that is paid is that the Z-axis linear expansion of the composite material can be huge, perhaps hundreds of parts per million. You'll have to consider carefully whether this might be a reliability concern, for example, this expansion might add stress to wirebonds.